christopher/oeis · Datasets at Hugging Face

a-number stringlengths 7 7	sequence sequencelengths 1 377	description stringlengths 3 852
A361332	[ "1", "2", "3", "4", "6", "5", "23", "7", "8", "9", "161", "10", "1771", "12", "11", "16", "23023", "13", "391391", "14", "15", "19", "7436429", "17", "18", "21", "20", "24" ]	Where n appears in A351495, or -1 if it never occurs.
A361333	[ "2", "3", "6", "23", "161", "1771", "23023", "391391", "7436429", "171037867", "4960098143" ]	Index of prime(n) in A351495.
A361334	[ "1", "2", "4", "7", "16", "26", "52", "100", "200", "394", "788", "1572", "3144", "6282", "12564", "25124", "50248", "100490", "200980", "401956", "803912", "1607818", "3215636", "6431268", "12862536", "25725066", "51450132", "102900260", "205800520", "411601034", "823202068", "1646404132", "3292808264", "6585616522", "13171233044" ]	Index of 2^n in A351495.
A361350	[ "11", "112", "1124", "11248", "1124816", "2486", "248620", "4860", "486018", "48601827", "4860182736", "486018273645", "8601827365", "860182736546", "86018273654656", "8601827365465667", "860182736546566780", "601273654656670", "60127365465667064", "-1273545704", "-127354570438", "-12735457043849", "-1273545704384962", "-127354570438496270", "1273545743849627", "127354574384962777", "12735457438496277791", "273545743849627779" ]	A variant of A359143 which includes the intermediate terms before digits are deleted (see Comments for precise definition).
A361351	[ "1", "1", "4", "7", "6", "5", "6", "3", "6", "9", "10000000000", "115502205511", "1440046600466", "19225142754633", "166668888866666", "1555555555555555", "16000880008800066", "194006440028800877", "1422046880284402844", "11116222228888849999", "600000000000000000000", "2600042000840006800021" ]	Carryless n-th powers of n base 10.
A361353	[ "1", "0", "1", "0", "1", "1", "0", "0", "5", "1", "0", "0", "15", "16", "1", "0", "0", "0", "175", "42", "1", "0", "0", "0", "735", "1225", "99", "1", "0", "0", "0", "0", "16065", "6769", "219", "1", "0", "0", "0", "0", "76545", "204400", "32830", "466", "1", "0", "0", "0", "0", "0", "2747745", "2001230", "147466", "968", "1", "0", "0", "0", "0", "0", "13835745", "56143395", "16813720", "632434", "1981", "1" ]	Triangle read by rows: T(n,k) is the number of simple quasi series-parallel matroids on [n] with rank k, 1 <= k <= n.
A361354	[ "1", "1", "2", "6", "32", "218", "2060", "23054", "314242", "4897410", "87427276", "1741312444", "38482278928", "931618115860", "24554678866736", "699328394272236", "21410158708401980", "701011980397033052", "24445424273647475096", "904440666571331841992", "35386719095200164370912", "1459756349974815778252152" ]	Number of simple quasi series-parallel matroids on [n].
A361355	[ "1", "0", "0", "0", "1", "0", "0", "0", "1", "0", "0", "0", "15", "1", "0", "0", "0", "0", "75", "1", "0", "0", "0", "0", "735", "280", "1", "0", "0", "0", "0", "0", "9345", "938", "1", "0", "0", "0", "0", "0", "76545", "77805", "2989", "1", "0", "0", "0", "0", "0", "0", "1865745", "536725", "9285", "1", "0", "0", "0", "0", "0", "0", "13835745", "27754650", "3334870", "28446", "1", "0" ]	Triangle read by rows: T(n,k) is the number of simple series-parallel matroids on [n] with rank k, 1 <= k <= n.
A361356	[ "1", "1", "3", "12", "55", "273", "1372", "6824", "33489", "162405", "779801", "3713436", "17560803", "82553597", "386105790", "1797803248", "8338313697", "38539754649", "177581276639", "815982230060", "3740047627071", "17103604731961", "78054858200448", "355541644914072", "1616688603539025" ]	Number of noncrossing caterpillars with n edges.
A361357	[ "1", "0", "1", "0", "0", "3", "0", "0", "4", "8", "0", "0", "5", "30", "20", "0", "0", "6", "75", "144", "48", "0", "0", "7", "154", "595", "504", "112", "0", "0", "8", "280", "1848", "2896", "1536", "256", "0", "0", "9", "468", "4788", "12060", "11268", "4320", "576", "0", "0", "10", "735", "10920", "40700", "58760", "38480", "11520", "1280" ]	Triangle read by rows: T(n,k) is the number of noncrossing caterpillars with n edges and diameter k, 0 <= k <= n.
A361359	[ "1", "1", "1", "4", "11", "49", "196", "868", "3721", "16306", "70891", "309739", "1350831", "5897934", "25740386", "112368153", "490489041", "2141121271", "9346382981", "40799215354", "178097506051", "777437032059", "3393689486976", "14814237183658", "64667544141561", "282288713218896", "1232255125682671" ]	Number of nonequivalent noncrossing caterpillars with n edges up to rotation.
A361360	[ "1", "1", "1", "3", "7", "28", "104", "448", "1886", "8212", "35556", "155124", "675897", "2950074", "12872294", "56188904", "245253691", "1070581703", "4673231521", "20399699635", "89048927767", "388718917440", "1696845506274", "7407120344070", "32333775400516", "141144364258374", "616127577376396" ]	Number of nonequivalent noncrossing caterpillars with n edges up to rotation and relection.
A361361	[ "1", "0", "0", "0", "1", "1", "1", "1", "1", "2", "2", "5", "5", "5", "2", "2", "6", "11", "33", "48", "66", "48", "33", "11", "6", "21", "68", "257", "556", "950", "1071", "950", "556", "257", "68", "21", "94", "510", "2443", "7126", "15393", "23644", "27606", "23644", "15393", "7126", "2443", "510", "94", "540", "4712", "27682", "102122", "270957", "526783", "781292", "887305", "781292", "526783", "270957", "102122", "27682", "4712", "540" ]	Triangle read by rows: T(n,k) is the number of bicolored cubic graphs on 2n unlabeled vertices with k vertices of the first color, n >= 0, 0 <= k <= 2*n.
A361362	[ "1", "0", "5", "23", "262", "4775", "126026", "4315481", "177939133", "8486268015", "457398466292", "27442206452816", "1812456359735759", "130630783430897459", "10200930403740584232", "857888417749736680977", "77299388952584465682198", "7429004444540543143978901", "758559920648248499878180973", "82006219796827162656265186759", "9357477001574426557631620060473" ]	Number of bicolored cubic graphs on 2n unlabeled vertices.
A361363	[ "1", "3", "8", "14", "15", "21", "26", "40", "130", "144", "182", "255", "310", "372", "465", "468", "680", "980", "1524", "2170", "2210", "2418", "2448", "4030", "4536", "7008", "7956", "8890", "9906", "10220", "10416", "10668", "12648", "16335", "16660", "17082", "20216", "24624", "30800", "36792", "41106", "44055", "48400", "65535", "77112", "78320", "85120", "97790", "143000", "149688" ]	Primitive terms of A259850.
A361364	[ "1", "10", "170", "6500", "332050", "19784060", "1296395700", "90616189800", "6637652225250", "503852804991500", "39337349077483420", "3142010167321271000", "255747325678297576100", "21150729618673827139000", "1773152567858996728205000", "150409554094012703302602000" ]	Number of 5-dimensional cubic lattice walks that start and end at origin after 2n steps, not touching origin at intermediate stages.
A361366	[ "1", "3", "16", "218", "9026", "907123" ]	Number of unlabeled simple planar digraphs with n nodes.
A361367	[ "7", "129", "7447", "1399245", "853468061", "1774125803324", "12983268697759210", "340896057593147232397", "32512334188761655225275067", "11365639780174824680535568799361", "14668665138188644335253106665956458513", "70315069858161131939222463684374769308619684" ]	Number of weakly 2-connected simple digraphs with n unlabeled nodes.
A361368	[ "2", "13", "199", "8782", "897604" ]	Number of weakly connected simple planar digraphs with n unlabeled nodes.
A361369	[ "7", "129", "6865", "774052" ]	Number of weakly 2-connected simple planar digraphs with n unlabeled nodes.
A361370	[ "42", "3270", "879508" ]	Number of weakly 3-connected simple digraphs with n unlabeled nodes.
A361371	[ "42", "2688", "316208" ]	Number of weakly 3-connected simple planar digraphs with n unlabeled nodes.
A361375	[ "1", "3", "21", "165", "1380", "11982", "106626", "965442", "8854725", "82022115", "765787773", "7195638909", "67973370618", "644991134880", "6143707229880", "58714212503784", "562741793028282", "5407273475087934", "52074626299010130", "502513862912425650", "4857975310180620720" ]	Expansion of 1/(1 - 9*x/(1 - x))^(1/3).
A361377	[ "1", "10", "3", "8", "5", "2", "7", "4", "9", "22", "19", "16", "33", "58", "13", "28", "25", "46", "21", "40", "17", "6", "23", "20", "39", "70", "43", "76", "47", "26", "11", "14", "29", "32", "15", "62", "37", "18", "35", "38", "63", "34", "59", "30", "53", "12", "31", "54", "85", "124", "51", "80", "83", "52", "49", "24", "77", "48", "119", "50", "27", "86", "55", "128", "89", "92" ]	Squares visited by a knight moving on a spirally numbered board always to the lowest unvisited coprime square.
A361379	[ "0", "1", "3", "2", "4", "6", "10", "12", "7", "15", "8", "16", "20", "24", "36", "40", "48", "5", "9", "11", "13", "19", "21", "25", "27", "43", "45", "51", "53", "14", "26", "28", "30", "54", "58", "60", "31", "63", "32", "64", "72", "80", "96", "136", "144", "160", "192", "17", "33", "35", "37", "41", "49", "67", "69", "73", "81", "83", "85", "97", "99", "101", "147", "149", "153" ]	Distinct values of A361401, in order of appearance.
A361381	[ "2", "4", "1", "2", "1", "4", "2", "1", "6", "2", "6", "4", "1", "1", "2", "8", "4", "4", "2", "1", "2", "2", "3", "2", "10", "12", "4", "2", "1", "4", "6", "7", "6", "3", "4", "1", "2", "10", "2", "6", "8", "7", "5", "2", "4", "4", "1", "2", "1", "10", "2", "5", "8", "4", "16", "4", "11", "1", "2", "12", "2", "9", "6", "5", "2", "6", "9", "6", "10", "10", "4", "1", "2", "12", "10", "3", "6", "4", "14", "9", "4", "18", "4", "4", "2", "1", "2", "3", "20", "10", "4", "5", "8", "10", "10", "18", "2", "22" ]	In continued fraction convergents of sqrt(d), where d=A005117(n) (squarefree numbers), the position of a/b where abs(a^2 - d*b^2) = 1 or 4.
A361382	[ "1", "2", "3", "6", "12", "20", "24", "60", "120", "120", "360", "720", "2520", "5040", "20160", "40320", "181440", "362880", "1814400", "3628800", "19958400", "39916800", "239500800", "479001600", "3113510400", "6227020800", "43589145600", "87178291200", "653837184000", "1307674368000", "10461394944000", "20922789888000" ]	The orders, with repetition, of subset-transitive permutation groups.
A361383	[ "1", "1", "2", "3", "3", "4", "5", "4", "7", "7", "7", "7", "8", "8", "8", "8", "8", "8", "8", "15", "16", "15", "16", "15", "18", "17", "18", "19", "19", "19", "19", "19", "19", "19", "19", "19", "19", "19", "19", "22", "24", "22", "24", "23", "24", "26", "26", "26", "26", "26", "26", "26", "26", "26", "29", "32", "33", "35", "32", "35", "32", "35", "32", "35", "32", "35", "32", "36", "35", "37" ]	a(n) is the number of locations 1..n-1 which can be reached starting from location i=a(n-1), where jumps from location i to i +- a(i) are permitted (within 1..n-1); a(1)=1. See example.
A361384	[ "0", "2", "2", "3", "3", "4", "4", "3", "4", "5", "4", "5", "4", "3", "4", "5", "4", "5", "4", "4", "3", "5", "4", "5", "5", "5", "5", "5", "5", "4", "5", "5", "4", "5", "5", "5", "5", "4", "4", "4", "5", "6", "5", "6", "5", "5", "6", "6", "5", "5", "6", "6", "5", "5", "6", "6", "6", "6", "5", "6", "5", "6", "6", "6", "5", "6", "5", "6", "5", "6", "5", "6", "4", "5", "6", "6", "6", "6", "5", "6", "5", "6", "6", "6", "6", "5", "6" ]	a(n) is the number of distinct prime factors of the n-th unitary harmonic number.
A361385	[ "0", "2", "2", "3", "3", "4", "4", "4", "5", "5", "5", "4", "3", "5", "5", "5", "4", "6", "5", "5", "6", "6", "5", "6", "5", "6", "6", "6", "5", "7", "4", "5", "5", "6", "7", "6", "6", "6", "7", "6", "6", "7", "6", "6", "6", "7", "6", "8", "7", "7", "7", "6", "7", "7", "7", "6", "8", "6", "5", "6", "7", "6", "7", "7", "6", "8", "7", "7", "8", "7", "6", "7", "8", "7", "6", "8", "7", "7", "7", "7", "9", "6", "8", "6", "8", "8", "7" ]	a(n) is the number of "Fermi-Dirac prime" factors (or I-components) of the n-th infinitary harmonic number.
A361386	[ "1", "3", "5", "6", "7", "9", "11", "12", "13", "14", "15", "17", "19", "21", "22", "23", "25", "27", "28", "29", "30", "31", "33", "35", "37", "38", "39", "41", "42", "43", "44", "45", "46", "47", "48", "49", "51", "53", "54", "55", "56", "57", "59", "60", "61", "62", "63", "65", "66", "67", "69", "70", "71", "73", "75", "76", "77", "78", "79", "81", "83", "84", "85", "86", "87", "89", "91" ]	Infinitary arithmetic numbers: numbers for which the arithmetic mean of the infinitary divisors is an integer.
A361387	[ "1", "6", "60", "270", "420", "630", "2970", "5460", "8190", "36720", "136500", "172900", "204750", "245700", "491400", "790398", "791700", "819000", "1037400", "1138320", "1187550", "1228500", "1801800", "2457000", "3767400", "4176900", "4504500", "5405400", "6397300", "6688500", "6741630", "7698600", "8353800", "10032750", "10228680" ]	Infinitary arithmetic numbers k whose mean infinitary divisor is an infinitary divisor of k.
A361388	[ "1", "2", "8", "96", "5376", "1981440", "5722536960", "138430238607360" ]	Number of orders of distances to vertices of n-dimensional cube.
A361389	[ "1", "3", "9", "696", "7656", "11880000000000", "16394400000000" ]	a(n) is the least positive integer that can be expressed as the sum of one or more consecutive nonzero palindromes in exactly n ways.
A361390	[ "1", "0", "1", "0", "1", "1", "0", "1", "2", "1", "0", "1", "4", "3", "1", "0", "1", "8", "9", "4", "1", "0", "1", "6", "7", "6", "5", "1", "0", "1", "2", "1", "4", "5", "6", "1", "0", "1", "4", "3", "6", "5", "6", "7", "1", "0", "1", "8", "9", "4", "5", "6", "9", "8", "1", "0", "1", "6", "7", "6", "5", "6", "3", "4", "9", "1", "0", "1", "2", "1", "4", "5", "6", "1", "2", "1", "10", "1", "0", "1", "4", "3", "6", "5", "6", "7", "6", "9", "100", "11", "1", "0", "1", "8", "9", "4", "5", "6", "9", "8", "1", "1000", "121", "12", "1" ]	Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) is carryless n^k base 10.
A361391	[ "1", "0", "0", "1", "0", "2", "0", "4", "2", "4", "5", "11", "0", "17", "15", "13", "15", "37", "18", "53", "24", "48", "78", "103", "23", "111", "152", "143", "123", "255", "110", "339", "238", "372", "495", "377", "243", "759", "845", "873", "414", "1259", "842", "1609", "1383", "1225", "2281", "2589", "1285", "2827", "2518", "3904", "3836", "5119", "3715", "4630" ]	Number of strict integer partitions of n with non-integer mean.
A361392	[ "0", "0", "0", "1", "0", "2", "1", "3", "2", "5", "4", "8", "7", "12", "12", "19", "19", "29", "31", "43", "48", "65", "73", "97", "110", "142", "164", "208", "240", "301", "350", "432", "504", "617", "719", "874", "1019", "1228", "1434", "1717", "2001", "2385", "2778", "3292", "3831", "4522", "5252", "6177", "7164", "8392", "9722", "11352", "13125", "15283", "17643" ]	Number of integer partitions of n whose first differences have mean -1.
A361393	[ "2", "3", "5", "6", "7", "10", "11", "12", "13", "14", "15", "17", "18", "19", "20", "21", "22", "23", "26", "28", "29", "30", "31", "33", "34", "35", "37", "38", "39", "41", "42", "43", "44", "45", "46", "47", "50", "51", "52", "53", "55", "57", "58", "59", "60", "61", "62", "63", "65", "66", "67", "68", "69", "70", "71", "73", "74", "75", "76", "77", "78", "79", "82", "83", "84", "85" ]	Positive integers k such that 2*omega(k) > bigomega(k).
A361394	[ "1", "1", "2", "2", "4", "6", "8", "11", "15", "20", "30", "38", "49", "65", "83", "108", "139", "178", "224", "286", "358", "437", "550", "684", "837", "1037", "1269", "1553", "1889", "2295", "2770", "3359", "4035", "4843", "5808", "6951", "8312", "9902", "11752", "13958", "16531", "19541", "23037", "27162", "31911", "37488", "43950", "51463", "60127", "70229" ]	Number of integer partitions of n where 2*(number of distinct parts) >= (number of parts).
A361395	[ "1", "2", "3", "4", "5", "6", "7", "9", "10", "11", "12", "13", "14", "15", "17", "18", "19", "20", "21", "22", "23", "24", "25", "26", "28", "29", "30", "31", "33", "34", "35", "36", "37", "38", "39", "40", "41", "42", "43", "44", "45", "46", "47", "49", "50", "51", "52", "53", "54", "55", "56", "57", "58", "59", "60", "61", "62", "63", "65", "66", "67", "68", "69", "70", "71", "73", "74" ]	Positive integers k such that 2*omega(k) >= bigomega(k).
A361396	[ "1", "2", "3", "4", "6", "7517", "15034", "18059", "22551", "28019", "30068", "30983", "36118", "45102", "56038", "61966", "65267", "67427", "67499", "71387", "84057", "84947", "90677", "92949", "97187", "112076", "115469", "123932", "127487", "130534", "130787", "134854", "134998", "142774", "168114", "169067", "169894", "181354", "185898", "191579", "194374", "195801" ]	Integers k such that 28phi(29197^3*k) is not a totient number where phi is the totient function.
A361397	[ "1", "1", "0", "1", "2", "0", "1", "4", "2", "0", "1", "6", "20", "4", "0", "1", "8", "54", "176", "10", "0", "1", "10", "104", "996", "1876", "28", "0", "1", "12", "170", "2944", "22734", "22064", "84", "0", "1", "14", "252", "6500", "108136", "577692", "275568", "264", "0", "1", "16", "350", "12144", "332050", "4525888", "15680628", "3584064", "858", "0" ]	Number A(n,k) of k-dimensional cubic lattice walks with 2n steps from origin to origin and avoiding early returns to the origin; square array A(n,k), n>=0, k>=0, read by antidiagonals.
A361398	[ "1", "2", "5", "3", "9", "12", "9", "4", "14", "28", "30", "21", "19", "21", "14", "5", "20", "53", "68", "60", "55", "74", "68", "32", "34", "60", "55", "36", "34", "32", "20", "6", "27", "89", "126", "134", "120", "181", "196", "108", "88", "181", "183", "136", "151", "164", "126", "45", "55", "134", "151", "129", "107", "136", "120", "54", "69", "108", "88", "54", "55", "45", "27" ]	An infiltration of two words, say x and y, is a shuffle of x and y optionally followed by replacements of pairs of consecutive equal symbols, say two d's, one of which comes from x and the other from y, by a single d (that cannot be part of another replacement); a(n) is the number of distinct infiltrations of the word given by the binary representation of n with itself.
A361399	[ "0", "1", "2", "1", "2", "5", "2", "3", "4", "5", "2", "5", "2", "5", "6", "3", "4", "9", "10", "5", "4", "5", "10", "11", "4", "5", "6", "5", "6", "13", "6", "7", "8", "9", "18", "9", "4", "9", "10", "11", "4", "9", "10", "5", "10", "5", "22", "11", "4", "9", "10", "5", "10", "5", "6", "11", "12", "13", "6", "13", "6", "13", "14", "7", "8", "17", "18", "9", "18", "9", "18", "19", "8", "9", "10", "19", "10", "21" ]	a(n) is the least k such that the binary expansion of n is a self-infiltration of that of k.
A361401	[ "0", "1", "3", "2", "4", "6", "10", "12", "3", "7", "15", "4", "8", "12", "16", "20", "24", "36", "40", "48", "5", "9", "11", "13", "19", "21", "25", "27", "43", "45", "51", "53", "6", "12", "14", "26", "28", "30", "54", "58", "60", "7", "15", "31", "63", "8", "16", "24", "32", "40", "48", "64", "72", "80", "96", "136", "144", "160", "192" ]	Irregular table T(n, k), n >= 0, k = 1..A361398(n); the n-th row lists the numbers whose binary expansion is a self-infiltration of that of n.
A361402	[ "5", "23", "599", "7899999999999999999999999999999999999999999999999899999999999999999" ]	a(1) = 5; a(n+1) is the smallest prime p > a(n) such that digsum(p) = a(n).
A361403	[ "1", "0", "5", "23", "247", "4660", "124480", "4286155", "177173770", "8460721770", "456369771864", "27394102475517", "1809905002448020", "130479709461582679", "10191059146232826353", "857183200472049855001", "77244717697104310952411", "7424434373914632379955822", "758150225111024064264853603", "81967014740890327829104517614", "9353488650500180241693235592248" ]	Number of bicolored connected cubic graphs on 2n unlabeled vertices.
A361404	[ "1", "1", "1", "2", "2", "2", "4", "6", "6", "4", "11", "20", "28", "20", "11", "34", "90", "148", "148", "90", "34", "156", "544", "1144", "1408", "1144", "544", "156", "1044", "5096", "13128", "20364", "20364", "13128", "5096", "1044", "12346", "79264", "250240", "472128", "580656", "472128", "250240", "79264", "12346" ]	Triangle read by rows: T(n,k) is the number of graphs with loops on n unlabeled vertices with k loops.
A361405	[ "1", "2", "28", "1408", "580656", "2658827456", "146702084635392", "98485306566812364032", "820443196111261227164076544", "86804253216450161933010414314819072", "119212631345634236227720012129209606659383296", "2166023316743980619769969171366251471253351621687457792" ]	Number of graphs with loops on 2n unlabeled vertices with n loops.
A361406	[ "1", "0", "1", "5", "63", "1052", "27336", "882321", "34455134", "1558650424", "80016369538", "4589908631503", "290839634055722", "20171917072658395", "1519875854413728667", "123616508830454828043", "10794216583730162449785", "1007179737486515827821590", "100007950522974604304016627", "10529173417583858651114779790", "1171605981584666223513790021758" ]	Number of bicolored connected cubic graphs on 2n unlabeled vertices with n vertices of each color.
A361407	[ "0", "1", "2", "10", "64", "490", "4595", "51063", "657623", "9592204", "155630924", "2771922417", "53673859357", "1121581872170", "25143397213226", "601751140758134", "15310778492310274", "412656423154230159", "11743600063060974656", "351882591907696156959" ]	Number of connected cubic graphs on 2n unlabeled vertices rooted at a vertex.
A361408	[ "0", "1", "5", "31", "248", "2382", "27233", "359800", "5364193", "88622485", "1602171855", "31410476113", "663240471075", "15001046054183", "361775504849332", "9266474332849318", "251217335356943672", "7186461542458525108", "216332059500870350414", "6835872042063656823802" ]	Number of connected cubic graphs on 2n unlabeled vertices rooted at a pair of indistinguishable vertices.
A361409	[ "1", "0", "1", "5", "66", "1071", "27606", "887305", "34583357", "1562797351", "80177945542", "4597212665432", "291214532031215", "20193430937073303", "1521240318892230748", "123711268485285686123", "10801367759750192440520", "1007762402877770768660697", "100058924666668698411972015", "10533938778032068908299390227", "1172080056205294525370971027435" ]	Number of bicolored cubic graphs on 2n unlabeled vertices with n vertices of each color.
A361410	[ "0", "1", "2", "11", "68", "510", "4712", "51877", "664520", "9662968", "156490473", "2783955994", "53863486240", "1124886942314", "25206326633070", "603048386506505", "15339533779133582", "413338072569232815", "11760801736217845686", "352342902996056683824" ]	Number of cubic graphs on 2n unlabeled vertices rooted at a vertex.
A361411	[ "0", "1", "5", "33", "257", "2443", "27682", "363759", "5405697", "89134360", "1609418390", "31525697245", "665263778962", "15039817276939", "362579178545598", "9284375250749758", "251643492565059981", "7197256536139662143", "216621907269166632361", "6844093745422473471562" ]	Number of cubic graphs on 2n unlabeled vertices rooted at a pair of indistinguishable vertices.
A361412	[ "1", "3", "12", "67", "441", "3464", "31616", "331997", "3961462", "53105424", "791237787", "12978022526", "232407307054", "4511887729886", "94385418177277", "2116529900006321", "50646269987874834", "1288091152941695791", "34697173459041347465", "986800102740080746702", "29548269236430810895013" ]	Number of connected 3-regular multigraphs on 2n unlabeled nodes rooted at an unoriented edge (or loop), loops allowed.
A361413	[ "0", "1", "1", "0", "1", "0", "1", "0", "0", "4128", "1", "10880", "641", "45904", "349496", "892088", "40873", "17695080" ]	Number of ways to tile an n X n square using rectangles with distinct dimensions where all the rectangle edge lengths are prime numbers.
A361414	[ "0", "0", "0", "0", "0", "1", "0", "2", "0", "1", "0", "2", "0", "1", "0", "7", "0", "2", "0", "2", "1", "1", "0", "6", "0", "1", "2", "1", "0", "1", "0", "33", "0", "1", "0", "4", "0", "1", "1", "5", "0", "2", "0", "1", "0", "1", "0", "23", "0", "2", "0", "2", "0", "6", "1", "5", "1", "1", "0", "3", "0", "1", "1", "200", "0", "1", "0", "2", "0", "1", "0", "19", "0", "1", "1", "1", "0", "2", "0", "24", "8", "1", "0", "3", "0" ]	Number of indecomposable non-abelian groups of order n.
A361418	[ "1", "4", "12", "16", "60", "36", "48", "256", "360", "4096", "180", "144", "240", "576", "768", "65536", "2520", "1048576", "12288", "900", "1260", "1296", "720", "2304", "1680", "9216", "2880", "5184", "3840", "147456", "196608", "36864", "27720", "46656", "3145728", "4398046511104", "61440", "3600", "6300", "18014398509481984", "10080", "20736" ]	a(n) is the least number with exactly n noninfinitary divisors.
A361419	[ "0", "6", "7", "9", "11", "18", "32", "44", "56", "62", "72", "82", "94", "96", "98", "102", "104", "110", "116", "122", "132", "136", "138", "146", "150", "152", "178", "180", "182", "210", "222", "226", "230", "236", "238", "242", "248", "252", "264", "272", "284", "292", "296", "304", "322", "332", "338", "342", "350", "356", "360", "374", "382", "390", "392", "404" ]	Numbers k such that there is a unique number m for which the sum of the aliquot infinitary divisors of m (A126168) is k.
A361420	[ "1", "6", "8", "15", "21", "52", "58", "82", "106", "118", "268", "158", "356", "1264", "1296", "388", "202", "214", "226", "130", "508", "524", "1936", "160", "138", "298", "692", "2608", "358", "3088", "288", "446", "454", "466", "932", "478", "432", "348", "1792", "538", "562", "578", "586", "12032", "1268", "748", "20736", "1348", "694", "706", "26368", "544", "758" ]	a(n) is the unique number m such that A126168(m) = A361419(n).
A361421	[ "840", "2040", "4440", "9240", "25320", "51000", "117480", "271320", "765480", "1531320", "3721800", "5956440", "12295560", "25086840", "54141960", "108284280", "250301640", "502213560", "1007626440", "2017856760", "4039750920", "8079502200", "19596145800", "44369345400", "71495068200", "115576350360", "231152701080" ]	Infinitary aliquot sequence starting at 840: a(1) = 840, a(n) = A126168(a(n-1)), for n >= 2.
A361422	[ "0", "1", "3", "2", "4", "17", "5", "8", "10", "18", "6", "19", "7", "20", "29", "9", "11", "47", "71", "21", "12", "22", "72", "96", "13", "23", "30", "24", "31", "121", "32", "36", "38", "48", "197", "49", "14", "50", "73", "97", "15", "51", "74", "25", "75", "26", "367", "98", "16", "52", "76", "27", "77", "28", "33", "99", "112", "122", "34", "123", "35", "124", "135", "37", "39" ]	Inverse permutation to A361379.
A361424	[ "1", "2", "2", "2", "6", "8", "4", "12", "48", "80", "4", "16", "80", "480", "1152", "8", "48", "480", "2880", "20160", "53760", "8", "53", "960", "13440", "107520" ]	Triangle read by rows: T(n,k) is the maximum of a certain measure of the difficulty level (see comments) for tiling an n X k rectangle with a set of integer-sided rectangular pieces, rounded down to the nearest integer.
A361425	[ "1", "2", "8", "80", "1152", "53760" ]	Maximum difficulty level (see A361424 for the definition) for tiling an n X n square with a set of integer-sided rectangles, rounded down to the nearest integer.
A361426	[ "2", "2", "6", "12", "16", "48", "53", "120", "320", "280", "1120", "2240", "2986", "8960", "17920", "26880", "53760", "107520", "134400", "268800", "537600", "591360", "1182720", "2365440", "2956800", "5677056", "11354112" ]	Maximum difficulty level (see A361424 for the definition) for tiling an n X 2 rectangle with a set of integer-sided rectangles, rounded down to the nearest integer.
A361427	[ "2", "6", "8", "48", "80", "480", "960", "1920", "3360", "13440", "20160", "60480", "80640", "201600", "967680", "1612800" ]	Maximum difficulty level (see A361424 for the definition) for tiling an n X 3 rectangle with a set of integer-sided rectangles, rounded down to the nearest integer.
A361428	[ "4", "12", "48", "80", "480", "2880", "13440", "53760", "107520", "322560", "725760" ]	Maximum difficulty level (see A361424 for the definition) for tiling an n X 4 rectangle with a set of integer-sided rectangles, rounded down to the nearest integer.
A361430	[ "1", "0", "0", "1", "0", "0", "0", "2", "1", "0", "0", "0", "0", "0", "0", "3", "0", "0", "0", "0", "0", "0", "0", "0", "1", "0", "2", "0", "0", "0", "0", "4", "0", "0", "0", "1", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "1", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "5", "0", "0", "0", "0", "0", "0", "0", "2", "0", "0", "0", "0", "0", "0", "0", "0", "3", "0", "0", "0", "0", "0", "0" ]	Multiplicative with a(p^e) = e - 1.
A361431	[ "1", "2", "24", "34802", "509145568", "142743029326162", "715761543475698773496", "63014651062141097287201438690", "96683719664587866428237173383906926464", "2573179910450886540215919614478751310457090316706", "1184101051443285881265166362742300236491599013268534224381864" ]	Number of ways to write n^2 as an ordered sum of n^2 squares of integers.
A361432	[ "1", "1", "0", "1", "1", "0", "1", "2", "2", "0", "1", "3", "6", "4", "0", "1", "4", "12", "20", "8", "0", "1", "5", "20", "54", "68", "16", "0", "1", "6", "30", "112", "252", "232", "32", "0", "1", "7", "42", "200", "656", "1188", "792", "64", "0", "1", "8", "56", "324", "1400", "3904", "5616", "2704", "128", "0", "1", "9", "72", "490", "2628", "10000", "23360", "26568", "9232", "256", "0" ]	Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..floor(n/2)} k^(n-j) * binomial(n,2*j).
A361433	[ "1", "0", "1", "1", "0", "1", "1", "1", "0", "1", "1", "0", "1", "1", "0", "1", "1", "1", "0", "1", "1", "0", "1", "1", "1", "0", "1", "1", "0", "1", "1", "0", "1", "1", "1", "0", "1", "1", "0", "1", "1", "1", "0", "1", "1", "0", "1", "1", "1", "0", "1", "1", "0", "1", "1", "0", "1", "1", "1", "0", "1", "1", "0", "1", "1", "1", "0", "1", "1", "0", "1", "1", "0", "1", "1", "1", "0", "1", "1", "0", "1", "1", "1", "0", "1", "1" ]	a(n) = number of squares in the n-th antidiagonal of the natural number array, A000027.
A361434	[ "1", "4", "11", "18", "59", "108", "187", "198", "274", "335", "338", "374", "381", "387", "433", "815", "848", "1495", "1629", "2002", "3554", "3565", "4112", "4318", "4569", "4592", "4613", "4618", "4643", "4727", "4733", "6103", "6118", "7074", "7153", "7319", "7521", "7562", "7567", "7684", "7748", "7757", "7764", "7989", "8205", "8561", "8620" ]	Positions in Pi where the leader in the race of digits changes.
A361435	[ "1", "3", "11", "34", "144", "165", "229", "517", "790", "6870", "12757", "21134", "54155", "226470", "193225", "431900", "948949", "3960994", "6674779", "7594013", "14204939", "32720909", "20369309", "176923605", "335119938" ]	a(n) is the least positive integer that can be expressed as the sum of one or more consecutive squarefree numbers in exactly n ways.
A361437	[ "2", "3", "4", "5", "6", "7", "8", "12", "15", "58", "59", "102", "111", "118", "164", "291", "589", "685", "1671", "1900", "1945", "4905" ]	Numbers k such that k! - Sum_{i=1..k-1} (-1)^(k-i)*i! is prime.
A361438	[ "1", "1", "3", "1", "7", "1", "3", "5", "15", "1", "31", "1", "3", "7", "9", "21", "63", "1", "127", "1", "3", "5", "15", "17", "51", "85", "255", "1", "7", "73", "511", "1", "3", "11", "31", "33", "93", "341", "1023", "1", "23", "89", "2047", "1", "3", "5", "7", "9", "13", "15", "21", "35", "39", "45", "63", "65", "91", "105", "117", "195", "273", "315", "455", "585", "819", "1365", "4095", "1", "8191", "1", "3", "43", "127", "129", "381", "5461", "16383" ]	Triangle T(n,k), n >= 1, 1 <= k <= A046801(n), read by rows, where T(n,k) is k-th smallest divisor of 2^n-1.
A361442	[ "0", "1", "-1", "2", "-3", "4", "3", "-5", "8", "-12", "5", "-8", "13", "-21", "33", "6", "-11", "19", "-32", "53", "-86", "-2", "-4", "15", "-34", "66", "-119", "205", "9", "-7", "11", "-26", "60", "-126", "245", "-450", "10", "-19", "26", "-37", "63", "-123", "249", "-494", "944", "7", "-17", "36", "-62", "99", "-162", "285", "-534", "1028", "-1972" ]	Infinite triangle T(n, k), n, k >= 0, read and filled by rows the greedy way with distinct integers such that for any n, k >= 0, T(n, k) + T(n+1, k) + T(n+1, k+1) = 0; each term is minimal in absolute value and in case of a tie, preference is given to the positive value.
A361443	[ "0", "1", "2", "3", "5", "6", "-2", "9", "10", "7", "16", "-10", "24", "14", "17", "22", "-13", "-29", "-16", "-18", "-25", "-24", "-20", "-27", "-35", "12", "-30", "-42", "-22", "-36", "-40", "-43", "-44", "-45", "-46", "21", "35", "28", "32", "38", "27", "37", "41", "30", "50", "46", "55", "51", "56", "39", "74", "54", "73", "67", "57", "78", "71", "59", "61", "80", "68", "79" ]	a(n) is the first term of the n-th row of A361442.
A361444	[ "1", "2", "3", "4", "7", "6", "5", "8", "9", "22", "141", "88", "111", "202", "55", "222", "11", "212", "99", "232", "121", "252", "101", "66", "131", "242", "191", "272", "77", "282", "151", "292", "171", "262", "181", "606", "313", "414", "343", "444", "353", "44", "303", "424", "33", "434", "323", "404", "383", "474", "535", "484", "373", "454", "333", "464", "363" ]	Lexicographically earliest sequence of distinct positive base-10 palindromes such that a(n) + a(n+1) is prime.
A361445	[ "3", "5", "7", "11", "13", "11", "13", "17", "31", "163", "229", "199", "313", "257", "277", "233", "223", "311", "331", "353", "373", "353", "167", "197", "373", "433", "463", "349", "359", "433", "443", "463", "433", "443", "787", "919", "727", "757", "787", "797", "397", "347", "727", "457", "467", "757", "727", "787", "857", "1009", "1019", "857" ]	Sums of consecutive terms of A361444.
A361446	[ "1", "3", "16", "99", "717", "5964", "56701", "611750", "7432491", "100838222", "1514749135", "24989362186", "449429188211", "8754181791029", "183621843677724", "4126714250580949", "98932328702693666", "2520187379996442269", "67980528958530199837", "1935753445850303203221", "58025998739501873764826" ]	Number of connected 3-regular multigraphs on 2n unlabeled nodes rooted at an oriented edge (or loop), loops allowed.
A361447	[ "1", "2", "9", "49", "338", "2744", "26025", "282419", "3463502", "47439030", "718618117", "11937743088", "215896959624", "4224096594516", "88919920910684", "2004237153640098", "48165411560792500", "1229462431057436457", "33221743136066636436", "947415638925100675208", "28436953641282225835143" ]	Number of connected 3-regular (cubic) multigraphs on 2n unlabeled nodes rooted at an unoriented edge (or loop) whose removal does not disconnect the graph, loops allowed.
A361448	[ "1", "2", "10", "66", "511", "4536", "45519", "512661", "6436571", "89505875", "1369509795", "22908806774", "416408493351", "8178599551905", "172690849144538", "3902128758180500", "93970611848528998", "2402929936231885063", "65029668312580777779", "1856984518220396165657", "55803367549204703645086" ]	Number of connected 3-regular multigraphs on 2n unlabeled nodes rooted at an oriented edge (or loop) whose removal does not disconnect the graph, loops allowed.
A361449	[ "1", "4", "1573", "235862938", "37155328943771767", "12458003910177278332403197817", "15868284521418341362691384074620547198698934", "126024243590219798408446284849897811759970155660106999854057796", "9633603531065043175094488158875624821526224424118142906010095879389674957042528276201" ]	Number of colorings of an n X n grid up to permutation of the colors with no element having the same color as any horizontal, diagonal or antidiagonal neighbor.
A361450	[ "1", "5", "2906", "656404264", "148049849095504726", "67939294184937980415465539016", "114130286115375064054502412158789812265958284", "1159829070306179232444894822978404171908276758235252386883985596", "110658909677185498376669680234621983460781735371211477687464832774947897935655888426146" ]	Number of colorings of an n X n grid up to permutation of the colors with no element having the same color as any horizontal or antidiagonal neighbor.
A361451	[ "1", "2", "716", "112073062", "18633407199331522", "6575857942770612176290018153", "8769438200005128572266011359369913757287151", "72530091349507692706447958441062812294511923156598114466468667", "5746371835090565784276352813398004749296101606959968049467898643632416711373273639694" ]	Number of colorings of an n X n grid up to permutation of the colors with no element having the same color as any horizontal, vertical or antidiagonal neighbor.
A361452	[ "1", "7", "4192", "953124784", "213291369981652792", "96638817185266245591837984336", "160065721141038888919235753368205172658011648", "1603869086916486859475402575499346988054543498175515730927380336", "150972529586126094166343144224892296826763766718771806614594599643773846828229334720096" ]	Number of colorings of an n X n grid up to permutation of the colors with no element having the same color as any diagonal or antidiagonal neighbor.
A361453	[ "1", "15", "4141", "450288795", "50602429743064097", "12123635532529660182357354372" ]	Number of colorings of the n X n knight graph up to permutation of the colors.
A361454	[ "1", "4", "17", "78", "360", "1835", "10168", "62271", "419701", "3107800", "25108419", "219982357", "2076785950", "21011123423", "226708386212", "2598075587529", "31509529248585", "403155101535686", "5426659537490872", "76655160760249052", "1133766220709242638", "17522418780011531368", "282452568669871514771", "4740645804610572971112" ]	Number of 4-regular multigraphs on n unlabeled nodes with 4 external legs, loops allowed.
A361455	[ "1", "0", "1", "0", "1", "3", "0", "18", "21", "25", "0", "1606", "1173", "774", "543", "0", "565080", "271790", "122595", "59830", "29281", "0", "734774776", "229224750", "70500705", "25349355", "10110735", "3781503", "0", "3523091615568", "685793359804", "138122171880", "35130437825", "11002159455", "3767987307", "1138779265" ]	Triangle read by rows: T(n,k) is the number of simple digraphs on labeled n nodes with k strongly connected components.
A361456	[ "1", "1", "3", "2", "13", "30", "24", "6", "75", "372", "780", "872", "546", "180", "24", "541", "4660", "18180", "42140", "64150", "66900", "48320", "23820", "7650", "1440", "120", "4683", "62130", "385980", "1487520", "3973770", "7789032", "11565360", "13238520", "11771130", "8124710", "4314420", "1729440", "506010", "101880", "12600", "720" ]	Irregular triangle read by rows. T(n,k) is the number of properly colored simple labeled graphs on [n] with exactly k edges, n >= 0, 0 <= k <= binomial(n,2).
A361457	[ "3", "4", "6", "7", "8", "10", "11", "12", "14", "15", "16", "17", "19", "20", "21", "23", "24", "26", "27", "28", "29", "30", "33", "34", "35", "36", "37", "38", "40" ]	Numbers k such that the first player has a winning strategy in the game described in the Comments.
A361460	[ "0", "1", "0", "0", "1", "0", "1", "1", "0", "0", "1", "0", "1", "0", "1", "0", "1", "0", "1", "1", "0", "0", "1", "0", "0", "1", "1", "0", "1", "0", "1", "0", "0", "1", "1", "0", "1", "0", "1", "0", "1", "0", "1", "1", "0", "0", "1", "1", "1", "0", "0", "0", "1", "1", "1", "1", "0", "0", "1", "0", "1", "1", "1", "0", "1", "0", "1", "0", "1", "0", "1", "0", "1", "1", "1", "1", "0", "0", "1", "1", "0", "0", "1", "1", "0", "0", "1", "0", "1", "0", "0", "1", "0", "1", "1", "0", "1", "1", "1", "0", "1", "0", "1", "1", "0", "0", "1", "0", "1", "1", "1", "0", "1", "0", "0", "1", "0", "1", "1", "1" ]	a(n) = 1 if A135504(n+1) = 2 * A135504(n), otherwise 0.
A361461	[ "2", "5", "7", "8", "11", "13", "15", "17", "19", "20", "23", "26", "27", "29", "31", "34", "35", "37", "39", "41", "43", "44", "47", "48", "49", "53", "54", "55", "56", "59", "61", "62", "63", "65", "67", "69", "71", "73", "74", "75", "76", "79", "80", "83", "84", "87", "89", "92", "94", "95", "97", "98", "99", "101", "103", "104", "107", "109", "110", "111", "113", "116", "118", "119", "120", "123", "124", "125", "127", "129", "131", "132" ]	Numbers k such that x(k+1) = 2 * x(k), when x(1)=1 and x(n) = x(n-1) + lcm(x(n-1),n), i.e., x(n) = A135504(n).
A361462	[ "2", "1", "2", "1", "1", "3", "1", "1", "1", "3", "1", "1", "1", "1", "1", "1", "1", "3", "1", "1", "3", "3", "1", "1", "1", "1", "1", "1", "1", "3", "1", "3", "1", "1", "1", "1", "1", "1", "1", "1", "1", "3", "1", "1", "3", "3", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "3", "1", "1", "1", "1", "1", "1", "1", "3", "1", "3", "1", "3", "1", "1", "1", "1", "1", "1", "1", "3", "1", "1", "1", "3", "1", "1", "3", "1", "1", "1", "1", "1", "3", "1", "3", "1", "1", "1", "1", "1", "1", "1", "1", "3", "1", "1", "1", "3", "1", "1", "1", "1", "1", "1", "1", "3", "1", "1", "3", "1", "1", "1" ]	a(n) = A135506(n) mod 4.
A361463	[ "0", "0", "0", "0", "0", "1", "0", "0", "0", "1", "0", "0", "0", "0", "0", "0", "0", "1", "0", "0", "1", "1", "0", "0", "0", "0", "0", "0", "0", "1", "0", "1", "0", "0", "0", "0", "0", "0", "0", "0", "0", "1", "0", "0", "1", "1", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "1", "0", "0", "0", "0", "0", "0", "0", "1", "0", "1", "0", "1", "0", "0", "0", "0", "0", "0", "0", "1", "0", "0", "0", "1", "0", "0", "1", "0", "0", "0", "0", "0", "1", "0", "1", "0", "0", "0", "0", "0", "0", "0", "0", "1", "0", "0", "0", "1", "0", "0", "0", "0", "0", "0", "0", "1", "0" ]	a(n) = 1 if A135506(n) == 3 (mod 4), otherwise 0.
A361464	[ "6", "10", "18", "21", "22", "30", "32", "42", "45", "46", "58", "66", "68", "70", "78", "82", "85", "91", "93", "102", "106", "114", "117", "126", "128", "130", "133", "138", "140", "141", "150", "162", "165", "166", "171", "176", "178", "187", "190", "198", "200", "205", "210", "212", "213", "214", "222", "226", "234", "235", "238", "248", "250", "253", "261", "262", "267", "270", "282", "294", "301", "306", "308", "310", "320" ]	Numbers k such that A135504(k+1) / A135504(k) is a multiple of 4.
A361465	[ "1", "0", "1", "0", "0", "1", "1", "0", "0", "0", "0", "0", "0", "0", "1", "0", "0", "0", "0", "0", "1", "0", "0", "0", "0", "0", "0", "1", "0", "0", "1", "0", "1", "0", "0", "0", "0", "0", "0", "0", "0", "1", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "1", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "1", "0", "0", "0", "0", "0", "0", "1", "0", "1", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "1" ]	a(n) = 1 if A017665(n) [the numerator of the sum of the reciprocals of the divisors of n] is a power of 2, otherwise 0.
A361466	[ "1", "1", "0", "0", "1", "1", "0", "0", "0", "1", "0", "0", "0", "1", "0", "0", "0", "0", "0", "0", "0", "0", "0", "1", "0", "0", "0", "0", "1", "1", "0", "0", "0", "0", "0", "0", "0", "1", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "1", "0", "0", "1", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "1", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "1", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "1", "0", "0", "0", "0", "0", "0", "0", "1", "0", "0", "1", "0", "0", "0", "0", "0", "0", "1" ]	a(n) = 1 if A017665(A003961(n)) is a power of 2, otherwise 0. Here A017665 is the numerator of the sum of the reciprocals of the divisors of n, and A003961 is fully multiplicative with a(p) = nextprime(p).
A361467	[ "1", "12", "30", "117", "56", "360", "132", "1080", "775", "672", "182", "3510", "306", "1584", "1680", "9801", "380", "9300", "552", "6552", "3960", "2184", "870", "32400", "2793", "3672", "19500", "15444", "992", "20160", "1406", "88452", "5460", "4560", "7392", "90675", "1722", "6624", "9180", "60480", "1892", "47520", "2256", "21294", "43400", "10440", "2862", "294030", "16093", "33516", "11400" ]	a(n) = A003961(n) * sigma(A003961(n)), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

A361332

[ "1", "2", "3", "4", "6", "5", "23", "7", "8", "9", "161", "10", "1771", "12", "11", "16", "23023", "13", "391391", "14", "15", "19", "7436429", "17", "18", "21", "20", "24" ]

Where n appears in A351495, or -1 if it never occurs.

A361333

[ "2", "3", "6", "23", "161", "1771", "23023", "391391", "7436429", "171037867", "4960098143" ]

Index of prime(n) in A351495.

A361334

[ "1", "2", "4", "7", "16", "26", "52", "100", "200", "394", "788", "1572", "3144", "6282", "12564", "25124", "50248", "100490", "200980", "401956", "803912", "1607818", "3215636", "6431268", "12862536", "25725066", "51450132", "102900260", "205800520", "411601034", "823202068", "1646404132", "3292808264", "6585616522", "13171233044" ]

Index of 2^n in A351495.

A361350

[ "11", "112", "1124", "11248", "1124816", "2486", "248620", "4860", "486018", "48601827", "4860182736", "486018273645", "8601827365", "860182736546", "86018273654656", "8601827365465667", "860182736546566780", "601273654656670", "60127365465667064", "-1273545704", "-127354570438", "-12735457043849", "-1273545704384962", "-127354570438496270", "1273545743849627", "127354574384962777", "12735457438496277791", "273545743849627779" ]

A variant of A359143 which includes the intermediate terms before digits are deleted (see Comments for precise definition).

A361351

[ "1", "1", "4", "7", "6", "5", "6", "3", "6", "9", "10000000000", "115502205511", "1440046600466", "19225142754633", "166668888866666", "1555555555555555", "16000880008800066", "194006440028800877", "1422046880284402844", "11116222228888849999", "600000000000000000000", "2600042000840006800021" ]

Carryless n-th powers of n base 10.

A361353

[ "1", "0", "1", "0", "1", "1", "0", "0", "5", "1", "0", "0", "15", "16", "1", "0", "0", "0", "175", "42", "1", "0", "0", "0", "735", "1225", "99", "1", "0", "0", "0", "0", "16065", "6769", "219", "1", "0", "0", "0", "0", "76545", "204400", "32830", "466", "1", "0", "0", "0", "0", "0", "2747745", "2001230", "147466", "968", "1", "0", "0", "0", "0", "0", "13835745", "56143395", "16813720", "632434", "1981", "1" ]

Triangle read by rows: T(n,k) is the number of simple quasi series-parallel matroids on [n] with rank k, 1 <= k <= n.

A361354

[ "1", "1", "2", "6", "32", "218", "2060", "23054", "314242", "4897410", "87427276", "1741312444", "38482278928", "931618115860", "24554678866736", "699328394272236", "21410158708401980", "701011980397033052", "24445424273647475096", "904440666571331841992", "35386719095200164370912", "1459756349974815778252152" ]

Number of simple quasi series-parallel matroids on [n].

A361355

[ "1", "0", "0", "0", "1", "0", "0", "0", "1", "0", "0", "0", "15", "1", "0", "0", "0", "0", "75", "1", "0", "0", "0", "0", "735", "280", "1", "0", "0", "0", "0", "0", "9345", "938", "1", "0", "0", "0", "0", "0", "76545", "77805", "2989", "1", "0", "0", "0", "0", "0", "0", "1865745", "536725", "9285", "1", "0", "0", "0", "0", "0", "0", "13835745", "27754650", "3334870", "28446", "1", "0" ]

Triangle read by rows: T(n,k) is the number of simple series-parallel matroids on [n] with rank k, 1 <= k <= n.

A361356

[ "1", "1", "3", "12", "55", "273", "1372", "6824", "33489", "162405", "779801", "3713436", "17560803", "82553597", "386105790", "1797803248", "8338313697", "38539754649", "177581276639", "815982230060", "3740047627071", "17103604731961", "78054858200448", "355541644914072", "1616688603539025" ]

Number of noncrossing caterpillars with n edges.

A361357

[ "1", "0", "1", "0", "0", "3", "0", "0", "4", "8", "0", "0", "5", "30", "20", "0", "0", "6", "75", "144", "48", "0", "0", "7", "154", "595", "504", "112", "0", "0", "8", "280", "1848", "2896", "1536", "256", "0", "0", "9", "468", "4788", "12060", "11268", "4320", "576", "0", "0", "10", "735", "10920", "40700", "58760", "38480", "11520", "1280" ]

Triangle read by rows: T(n,k) is the number of noncrossing caterpillars with n edges and diameter k, 0 <= k <= n.

A361359

[ "1", "1", "1", "4", "11", "49", "196", "868", "3721", "16306", "70891", "309739", "1350831", "5897934", "25740386", "112368153", "490489041", "2141121271", "9346382981", "40799215354", "178097506051", "777437032059", "3393689486976", "14814237183658", "64667544141561", "282288713218896", "1232255125682671" ]

Number of nonequivalent noncrossing caterpillars with n edges up to rotation.

A361360

[ "1", "1", "1", "3", "7", "28", "104", "448", "1886", "8212", "35556", "155124", "675897", "2950074", "12872294", "56188904", "245253691", "1070581703", "4673231521", "20399699635", "89048927767", "388718917440", "1696845506274", "7407120344070", "32333775400516", "141144364258374", "616127577376396" ]

Number of nonequivalent noncrossing caterpillars with n edges up to rotation and relection.

A361361

[ "1", "0", "0", "0", "1", "1", "1", "1", "1", "2", "2", "5", "5", "5", "2", "2", "6", "11", "33", "48", "66", "48", "33", "11", "6", "21", "68", "257", "556", "950", "1071", "950", "556", "257", "68", "21", "94", "510", "2443", "7126", "15393", "23644", "27606", "23644", "15393", "7126", "2443", "510", "94", "540", "4712", "27682", "102122", "270957", "526783", "781292", "887305", "781292", "526783", "270957", "102122", "27682", "4712", "540" ]

Triangle read by rows: T(n,k) is the number of bicolored cubic graphs on 2n unlabeled vertices with k vertices of the first color, n >= 0, 0 <= k <= 2*n.

A361362

[ "1", "0", "5", "23", "262", "4775", "126026", "4315481", "177939133", "8486268015", "457398466292", "27442206452816", "1812456359735759", "130630783430897459", "10200930403740584232", "857888417749736680977", "77299388952584465682198", "7429004444540543143978901", "758559920648248499878180973", "82006219796827162656265186759", "9357477001574426557631620060473" ]

Number of bicolored cubic graphs on 2n unlabeled vertices.

A361363

[ "1", "3", "8", "14", "15", "21", "26", "40", "130", "144", "182", "255", "310", "372", "465", "468", "680", "980", "1524", "2170", "2210", "2418", "2448", "4030", "4536", "7008", "7956", "8890", "9906", "10220", "10416", "10668", "12648", "16335", "16660", "17082", "20216", "24624", "30800", "36792", "41106", "44055", "48400", "65535", "77112", "78320", "85120", "97790", "143000", "149688" ]

Primitive terms of A259850.

A361364

[ "1", "10", "170", "6500", "332050", "19784060", "1296395700", "90616189800", "6637652225250", "503852804991500", "39337349077483420", "3142010167321271000", "255747325678297576100", "21150729618673827139000", "1773152567858996728205000", "150409554094012703302602000" ]

Number of 5-dimensional cubic lattice walks that start and end at origin after 2n steps, not touching origin at intermediate stages.

A361366

[ "1", "3", "16", "218", "9026", "907123" ]

Number of unlabeled simple planar digraphs with n nodes.

A361367

[ "7", "129", "7447", "1399245", "853468061", "1774125803324", "12983268697759210", "340896057593147232397", "32512334188761655225275067", "11365639780174824680535568799361", "14668665138188644335253106665956458513", "70315069858161131939222463684374769308619684" ]

Number of weakly 2-connected simple digraphs with n unlabeled nodes.

A361368

[ "2", "13", "199", "8782", "897604" ]

Number of weakly connected simple planar digraphs with n unlabeled nodes.

A361369

[ "7", "129", "6865", "774052" ]

Number of weakly 2-connected simple planar digraphs with n unlabeled nodes.

A361370

[ "42", "3270", "879508" ]

Number of weakly 3-connected simple digraphs with n unlabeled nodes.

A361371

[ "42", "2688", "316208" ]

Number of weakly 3-connected simple planar digraphs with n unlabeled nodes.

A361375

[ "1", "3", "21", "165", "1380", "11982", "106626", "965442", "8854725", "82022115", "765787773", "7195638909", "67973370618", "644991134880", "6143707229880", "58714212503784", "562741793028282", "5407273475087934", "52074626299010130", "502513862912425650", "4857975310180620720" ]

Expansion of 1/(1 - 9*x/(1 - x))^(1/3).

A361377

[ "1", "10", "3", "8", "5", "2", "7", "4", "9", "22", "19", "16", "33", "58", "13", "28", "25", "46", "21", "40", "17", "6", "23", "20", "39", "70", "43", "76", "47", "26", "11", "14", "29", "32", "15", "62", "37", "18", "35", "38", "63", "34", "59", "30", "53", "12", "31", "54", "85", "124", "51", "80", "83", "52", "49", "24", "77", "48", "119", "50", "27", "86", "55", "128", "89", "92" ]

Squares visited by a knight moving on a spirally numbered board always to the lowest unvisited coprime square.

A361379

[ "0", "1", "3", "2", "4", "6", "10", "12", "7", "15", "8", "16", "20", "24", "36", "40", "48", "5", "9", "11", "13", "19", "21", "25", "27", "43", "45", "51", "53", "14", "26", "28", "30", "54", "58", "60", "31", "63", "32", "64", "72", "80", "96", "136", "144", "160", "192", "17", "33", "35", "37", "41", "49", "67", "69", "73", "81", "83", "85", "97", "99", "101", "147", "149", "153" ]

Distinct values of A361401, in order of appearance.

A361381

[ "2", "4", "1", "2", "1", "4", "2", "1", "6", "2", "6", "4", "1", "1", "2", "8", "4", "4", "2", "1", "2", "2", "3", "2", "10", "12", "4", "2", "1", "4", "6", "7", "6", "3", "4", "1", "2", "10", "2", "6", "8", "7", "5", "2", "4", "4", "1", "2", "1", "10", "2", "5", "8", "4", "16", "4", "11", "1", "2", "12", "2", "9", "6", "5", "2", "6", "9", "6", "10", "10", "4", "1", "2", "12", "10", "3", "6", "4", "14", "9", "4", "18", "4", "4", "2", "1", "2", "3", "20", "10", "4", "5", "8", "10", "10", "18", "2", "22" ]

In continued fraction convergents of sqrt(d), where d=A005117(n) (squarefree numbers), the position of a/b where abs(a^2 - d*b^2) = 1 or 4.

A361382

[ "1", "2", "3", "6", "12", "20", "24", "60", "120", "120", "360", "720", "2520", "5040", "20160", "40320", "181440", "362880", "1814400", "3628800", "19958400", "39916800", "239500800", "479001600", "3113510400", "6227020800", "43589145600", "87178291200", "653837184000", "1307674368000", "10461394944000", "20922789888000" ]

The orders, with repetition, of subset-transitive permutation groups.

A361383

[ "1", "1", "2", "3", "3", "4", "5", "4", "7", "7", "7", "7", "8", "8", "8", "8", "8", "8", "8", "15", "16", "15", "16", "15", "18", "17", "18", "19", "19", "19", "19", "19", "19", "19", "19", "19", "19", "19", "19", "22", "24", "22", "24", "23", "24", "26", "26", "26", "26", "26", "26", "26", "26", "26", "29", "32", "33", "35", "32", "35", "32", "35", "32", "35", "32", "35", "32", "36", "35", "37" ]

a(n) is the number of locations 1..n-1 which can be reached starting from location i=a(n-1), where jumps from location i to i +- a(i) are permitted (within 1..n-1); a(1)=1. See example.

A361384

[ "0", "2", "2", "3", "3", "4", "4", "3", "4", "5", "4", "5", "4", "3", "4", "5", "4", "5", "4", "4", "3", "5", "4", "5", "5", "5", "5", "5", "5", "4", "5", "5", "4", "5", "5", "5", "5", "4", "4", "4", "5", "6", "5", "6", "5", "5", "6", "6", "5", "5", "6", "6", "5", "5", "6", "6", "6", "6", "5", "6", "5", "6", "6", "6", "5", "6", "5", "6", "5", "6", "5", "6", "4", "5", "6", "6", "6", "6", "5", "6", "5", "6", "6", "6", "6", "5", "6" ]

a(n) is the number of distinct prime factors of the n-th unitary harmonic number.

A361385

[ "0", "2", "2", "3", "3", "4", "4", "4", "5", "5", "5", "4", "3", "5", "5", "5", "4", "6", "5", "5", "6", "6", "5", "6", "5", "6", "6", "6", "5", "7", "4", "5", "5", "6", "7", "6", "6", "6", "7", "6", "6", "7", "6", "6", "6", "7", "6", "8", "7", "7", "7", "6", "7", "7", "7", "6", "8", "6", "5", "6", "7", "6", "7", "7", "6", "8", "7", "7", "8", "7", "6", "7", "8", "7", "6", "8", "7", "7", "7", "7", "9", "6", "8", "6", "8", "8", "7" ]

a(n) is the number of "Fermi-Dirac prime" factors (or I-components) of the n-th infinitary harmonic number.

A361386

[ "1", "3", "5", "6", "7", "9", "11", "12", "13", "14", "15", "17", "19", "21", "22", "23", "25", "27", "28", "29", "30", "31", "33", "35", "37", "38", "39", "41", "42", "43", "44", "45", "46", "47", "48", "49", "51", "53", "54", "55", "56", "57", "59", "60", "61", "62", "63", "65", "66", "67", "69", "70", "71", "73", "75", "76", "77", "78", "79", "81", "83", "84", "85", "86", "87", "89", "91" ]

Infinitary arithmetic numbers: numbers for which the arithmetic mean of the infinitary divisors is an integer.

A361387

[ "1", "6", "60", "270", "420", "630", "2970", "5460", "8190", "36720", "136500", "172900", "204750", "245700", "491400", "790398", "791700", "819000", "1037400", "1138320", "1187550", "1228500", "1801800", "2457000", "3767400", "4176900", "4504500", "5405400", "6397300", "6688500", "6741630", "7698600", "8353800", "10032750", "10228680" ]

Infinitary arithmetic numbers k whose mean infinitary divisor is an infinitary divisor of k.

A361388

[ "1", "2", "8", "96", "5376", "1981440", "5722536960", "138430238607360" ]

Number of orders of distances to vertices of n-dimensional cube.

A361389

[ "1", "3", "9", "696", "7656", "11880000000000", "16394400000000" ]

a(n) is the least positive integer that can be expressed as the sum of one or more consecutive nonzero palindromes in exactly n ways.

A361390

[ "1", "0", "1", "0", "1", "1", "0", "1", "2", "1", "0", "1", "4", "3", "1", "0", "1", "8", "9", "4", "1", "0", "1", "6", "7", "6", "5", "1", "0", "1", "2", "1", "4", "5", "6", "1", "0", "1", "4", "3", "6", "5", "6", "7", "1", "0", "1", "8", "9", "4", "5", "6", "9", "8", "1", "0", "1", "6", "7", "6", "5", "6", "3", "4", "9", "1", "0", "1", "2", "1", "4", "5", "6", "1", "2", "1", "10", "1", "0", "1", "4", "3", "6", "5", "6", "7", "6", "9", "100", "11", "1", "0", "1", "8", "9", "4", "5", "6", "9", "8", "1", "1000", "121", "12", "1" ]

Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) is carryless n^k base 10.

A361391

[ "1", "0", "0", "1", "0", "2", "0", "4", "2", "4", "5", "11", "0", "17", "15", "13", "15", "37", "18", "53", "24", "48", "78", "103", "23", "111", "152", "143", "123", "255", "110", "339", "238", "372", "495", "377", "243", "759", "845", "873", "414", "1259", "842", "1609", "1383", "1225", "2281", "2589", "1285", "2827", "2518", "3904", "3836", "5119", "3715", "4630" ]

Number of strict integer partitions of n with non-integer mean.

A361392

[ "0", "0", "0", "1", "0", "2", "1", "3", "2", "5", "4", "8", "7", "12", "12", "19", "19", "29", "31", "43", "48", "65", "73", "97", "110", "142", "164", "208", "240", "301", "350", "432", "504", "617", "719", "874", "1019", "1228", "1434", "1717", "2001", "2385", "2778", "3292", "3831", "4522", "5252", "6177", "7164", "8392", "9722", "11352", "13125", "15283", "17643" ]

Number of integer partitions of n whose first differences have mean -1.

A361393

[ "2", "3", "5", "6", "7", "10", "11", "12", "13", "14", "15", "17", "18", "19", "20", "21", "22", "23", "26", "28", "29", "30", "31", "33", "34", "35", "37", "38", "39", "41", "42", "43", "44", "45", "46", "47", "50", "51", "52", "53", "55", "57", "58", "59", "60", "61", "62", "63", "65", "66", "67", "68", "69", "70", "71", "73", "74", "75", "76", "77", "78", "79", "82", "83", "84", "85" ]

Positive integers k such that 2*omega(k) > bigomega(k).

A361394

[ "1", "1", "2", "2", "4", "6", "8", "11", "15", "20", "30", "38", "49", "65", "83", "108", "139", "178", "224", "286", "358", "437", "550", "684", "837", "1037", "1269", "1553", "1889", "2295", "2770", "3359", "4035", "4843", "5808", "6951", "8312", "9902", "11752", "13958", "16531", "19541", "23037", "27162", "31911", "37488", "43950", "51463", "60127", "70229" ]

Number of integer partitions of n where 2*(number of distinct parts) >= (number of parts).

A361395

[ "1", "2", "3", "4", "5", "6", "7", "9", "10", "11", "12", "13", "14", "15", "17", "18", "19", "20", "21", "22", "23", "24", "25", "26", "28", "29", "30", "31", "33", "34", "35", "36", "37", "38", "39", "40", "41", "42", "43", "44", "45", "46", "47", "49", "50", "51", "52", "53", "54", "55", "56", "57", "58", "59", "60", "61", "62", "63", "65", "66", "67", "68", "69", "70", "71", "73", "74" ]

Positive integers k such that 2*omega(k) >= bigomega(k).

A361396

[ "1", "2", "3", "4", "6", "7517", "15034", "18059", "22551", "28019", "30068", "30983", "36118", "45102", "56038", "61966", "65267", "67427", "67499", "71387", "84057", "84947", "90677", "92949", "97187", "112076", "115469", "123932", "127487", "130534", "130787", "134854", "134998", "142774", "168114", "169067", "169894", "181354", "185898", "191579", "194374", "195801" ]

Integers k such that 28*phi(29*197^3*k) is not a totient number where phi is the totient function.

A361397

[ "1", "1", "0", "1", "2", "0", "1", "4", "2", "0", "1", "6", "20", "4", "0", "1", "8", "54", "176", "10", "0", "1", "10", "104", "996", "1876", "28", "0", "1", "12", "170", "2944", "22734", "22064", "84", "0", "1", "14", "252", "6500", "108136", "577692", "275568", "264", "0", "1", "16", "350", "12144", "332050", "4525888", "15680628", "3584064", "858", "0" ]

Number A(n,k) of k-dimensional cubic lattice walks with 2n steps from origin to origin and avoiding early returns to the origin; square array A(n,k), n>=0, k>=0, read by antidiagonals.

A361398

[ "1", "2", "5", "3", "9", "12", "9", "4", "14", "28", "30", "21", "19", "21", "14", "5", "20", "53", "68", "60", "55", "74", "68", "32", "34", "60", "55", "36", "34", "32", "20", "6", "27", "89", "126", "134", "120", "181", "196", "108", "88", "181", "183", "136", "151", "164", "126", "45", "55", "134", "151", "129", "107", "136", "120", "54", "69", "108", "88", "54", "55", "45", "27" ]

An infiltration of two words, say x and y, is a shuffle of x and y optionally followed by replacements of pairs of consecutive equal symbols, say two d's, one of which comes from x and the other from y, by a single d (that cannot be part of another replacement); a(n) is the number of distinct infiltrations of the word given by the binary representation of n with itself.

A361399

[ "0", "1", "2", "1", "2", "5", "2", "3", "4", "5", "2", "5", "2", "5", "6", "3", "4", "9", "10", "5", "4", "5", "10", "11", "4", "5", "6", "5", "6", "13", "6", "7", "8", "9", "18", "9", "4", "9", "10", "11", "4", "9", "10", "5", "10", "5", "22", "11", "4", "9", "10", "5", "10", "5", "6", "11", "12", "13", "6", "13", "6", "13", "14", "7", "8", "17", "18", "9", "18", "9", "18", "19", "8", "9", "10", "19", "10", "21" ]

a(n) is the least k such that the binary expansion of n is a self-infiltration of that of k.

A361401

[ "0", "1", "3", "2", "4", "6", "10", "12", "3", "7", "15", "4", "8", "12", "16", "20", "24", "36", "40", "48", "5", "9", "11", "13", "19", "21", "25", "27", "43", "45", "51", "53", "6", "12", "14", "26", "28", "30", "54", "58", "60", "7", "15", "31", "63", "8", "16", "24", "32", "40", "48", "64", "72", "80", "96", "136", "144", "160", "192" ]

Irregular table T(n, k), n >= 0, k = 1..A361398(n); the n-th row lists the numbers whose binary expansion is a self-infiltration of that of n.

A361402

[ "5", "23", "599", "7899999999999999999999999999999999999999999999999899999999999999999" ]

a(1) = 5; a(n+1) is the smallest prime p > a(n) such that digsum(p) = a(n).

A361403

[ "1", "0", "5", "23", "247", "4660", "124480", "4286155", "177173770", "8460721770", "456369771864", "27394102475517", "1809905002448020", "130479709461582679", "10191059146232826353", "857183200472049855001", "77244717697104310952411", "7424434373914632379955822", "758150225111024064264853603", "81967014740890327829104517614", "9353488650500180241693235592248" ]

Number of bicolored connected cubic graphs on 2n unlabeled vertices.

A361404

[ "1", "1", "1", "2", "2", "2", "4", "6", "6", "4", "11", "20", "28", "20", "11", "34", "90", "148", "148", "90", "34", "156", "544", "1144", "1408", "1144", "544", "156", "1044", "5096", "13128", "20364", "20364", "13128", "5096", "1044", "12346", "79264", "250240", "472128", "580656", "472128", "250240", "79264", "12346" ]

Triangle read by rows: T(n,k) is the number of graphs with loops on n unlabeled vertices with k loops.

A361405

[ "1", "2", "28", "1408", "580656", "2658827456", "146702084635392", "98485306566812364032", "820443196111261227164076544", "86804253216450161933010414314819072", "119212631345634236227720012129209606659383296", "2166023316743980619769969171366251471253351621687457792" ]

Number of graphs with loops on 2n unlabeled vertices with n loops.

A361406

[ "1", "0", "1", "5", "63", "1052", "27336", "882321", "34455134", "1558650424", "80016369538", "4589908631503", "290839634055722", "20171917072658395", "1519875854413728667", "123616508830454828043", "10794216583730162449785", "1007179737486515827821590", "100007950522974604304016627", "10529173417583858651114779790", "1171605981584666223513790021758" ]

Number of bicolored connected cubic graphs on 2n unlabeled vertices with n vertices of each color.

A361407

[ "0", "1", "2", "10", "64", "490", "4595", "51063", "657623", "9592204", "155630924", "2771922417", "53673859357", "1121581872170", "25143397213226", "601751140758134", "15310778492310274", "412656423154230159", "11743600063060974656", "351882591907696156959" ]

Number of connected cubic graphs on 2n unlabeled vertices rooted at a vertex.

A361408

[ "0", "1", "5", "31", "248", "2382", "27233", "359800", "5364193", "88622485", "1602171855", "31410476113", "663240471075", "15001046054183", "361775504849332", "9266474332849318", "251217335356943672", "7186461542458525108", "216332059500870350414", "6835872042063656823802" ]

Number of connected cubic graphs on 2n unlabeled vertices rooted at a pair of indistinguishable vertices.

A361409

[ "1", "0", "1", "5", "66", "1071", "27606", "887305", "34583357", "1562797351", "80177945542", "4597212665432", "291214532031215", "20193430937073303", "1521240318892230748", "123711268485285686123", "10801367759750192440520", "1007762402877770768660697", "100058924666668698411972015", "10533938778032068908299390227", "1172080056205294525370971027435" ]

Number of bicolored cubic graphs on 2n unlabeled vertices with n vertices of each color.

A361410

[ "0", "1", "2", "11", "68", "510", "4712", "51877", "664520", "9662968", "156490473", "2783955994", "53863486240", "1124886942314", "25206326633070", "603048386506505", "15339533779133582", "413338072569232815", "11760801736217845686", "352342902996056683824" ]

Number of cubic graphs on 2n unlabeled vertices rooted at a vertex.

A361411

[ "0", "1", "5", "33", "257", "2443", "27682", "363759", "5405697", "89134360", "1609418390", "31525697245", "665263778962", "15039817276939", "362579178545598", "9284375250749758", "251643492565059981", "7197256536139662143", "216621907269166632361", "6844093745422473471562" ]

Number of cubic graphs on 2n unlabeled vertices rooted at a pair of indistinguishable vertices.

A361412

[ "1", "3", "12", "67", "441", "3464", "31616", "331997", "3961462", "53105424", "791237787", "12978022526", "232407307054", "4511887729886", "94385418177277", "2116529900006321", "50646269987874834", "1288091152941695791", "34697173459041347465", "986800102740080746702", "29548269236430810895013" ]

Number of connected 3-regular multigraphs on 2n unlabeled nodes rooted at an unoriented edge (or loop), loops allowed.

A361413

[ "0", "1", "1", "0", "1", "0", "1", "0", "0", "4128", "1", "10880", "641", "45904", "349496", "892088", "40873", "17695080" ]

Number of ways to tile an n X n square using rectangles with distinct dimensions where all the rectangle edge lengths are prime numbers.

A361414

[ "0", "0", "0", "0", "0", "1", "0", "2", "0", "1", "0", "2", "0", "1", "0", "7", "0", "2", "0", "2", "1", "1", "0", "6", "0", "1", "2", "1", "0", "1", "0", "33", "0", "1", "0", "4", "0", "1", "1", "5", "0", "2", "0", "1", "0", "1", "0", "23", "0", "2", "0", "2", "0", "6", "1", "5", "1", "1", "0", "3", "0", "1", "1", "200", "0", "1", "0", "2", "0", "1", "0", "19", "0", "1", "1", "1", "0", "2", "0", "24", "8", "1", "0", "3", "0" ]

Number of indecomposable non-abelian groups of order n.

A361418

[ "1", "4", "12", "16", "60", "36", "48", "256", "360", "4096", "180", "144", "240", "576", "768", "65536", "2520", "1048576", "12288", "900", "1260", "1296", "720", "2304", "1680", "9216", "2880", "5184", "3840", "147456", "196608", "36864", "27720", "46656", "3145728", "4398046511104", "61440", "3600", "6300", "18014398509481984", "10080", "20736" ]

a(n) is the least number with exactly n noninfinitary divisors.

A361419

[ "0", "6", "7", "9", "11", "18", "32", "44", "56", "62", "72", "82", "94", "96", "98", "102", "104", "110", "116", "122", "132", "136", "138", "146", "150", "152", "178", "180", "182", "210", "222", "226", "230", "236", "238", "242", "248", "252", "264", "272", "284", "292", "296", "304", "322", "332", "338", "342", "350", "356", "360", "374", "382", "390", "392", "404" ]

Numbers k such that there is a unique number m for which the sum of the aliquot infinitary divisors of m (A126168) is k.

A361420

[ "1", "6", "8", "15", "21", "52", "58", "82", "106", "118", "268", "158", "356", "1264", "1296", "388", "202", "214", "226", "130", "508", "524", "1936", "160", "138", "298", "692", "2608", "358", "3088", "288", "446", "454", "466", "932", "478", "432", "348", "1792", "538", "562", "578", "586", "12032", "1268", "748", "20736", "1348", "694", "706", "26368", "544", "758" ]

a(n) is the unique number m such that A126168(m) = A361419(n).

A361421

[ "840", "2040", "4440", "9240", "25320", "51000", "117480", "271320", "765480", "1531320", "3721800", "5956440", "12295560", "25086840", "54141960", "108284280", "250301640", "502213560", "1007626440", "2017856760", "4039750920", "8079502200", "19596145800", "44369345400", "71495068200", "115576350360", "231152701080" ]

Infinitary aliquot sequence starting at 840: a(1) = 840, a(n) = A126168(a(n-1)), for n >= 2.

A361422

[ "0", "1", "3", "2", "4", "17", "5", "8", "10", "18", "6", "19", "7", "20", "29", "9", "11", "47", "71", "21", "12", "22", "72", "96", "13", "23", "30", "24", "31", "121", "32", "36", "38", "48", "197", "49", "14", "50", "73", "97", "15", "51", "74", "25", "75", "26", "367", "98", "16", "52", "76", "27", "77", "28", "33", "99", "112", "122", "34", "123", "35", "124", "135", "37", "39" ]

Inverse permutation to A361379.

A361424

[ "1", "2", "2", "2", "6", "8", "4", "12", "48", "80", "4", "16", "80", "480", "1152", "8", "48", "480", "2880", "20160", "53760", "8", "53", "960", "13440", "107520" ]

Triangle read by rows: T(n,k) is the maximum of a certain measure of the difficulty level (see comments) for tiling an n X k rectangle with a set of integer-sided rectangular pieces, rounded down to the nearest integer.

A361425

[ "1", "2", "8", "80", "1152", "53760" ]

Maximum difficulty level (see A361424 for the definition) for tiling an n X n square with a set of integer-sided rectangles, rounded down to the nearest integer.

A361426

[ "2", "2", "6", "12", "16", "48", "53", "120", "320", "280", "1120", "2240", "2986", "8960", "17920", "26880", "53760", "107520", "134400", "268800", "537600", "591360", "1182720", "2365440", "2956800", "5677056", "11354112" ]

Maximum difficulty level (see A361424 for the definition) for tiling an n X 2 rectangle with a set of integer-sided rectangles, rounded down to the nearest integer.

A361427

[ "2", "6", "8", "48", "80", "480", "960", "1920", "3360", "13440", "20160", "60480", "80640", "201600", "967680", "1612800" ]

Maximum difficulty level (see A361424 for the definition) for tiling an n X 3 rectangle with a set of integer-sided rectangles, rounded down to the nearest integer.

A361428

[ "4", "12", "48", "80", "480", "2880", "13440", "53760", "107520", "322560", "725760" ]

Maximum difficulty level (see A361424 for the definition) for tiling an n X 4 rectangle with a set of integer-sided rectangles, rounded down to the nearest integer.

A361430

[ "1", "0", "0", "1", "0", "0", "0", "2", "1", "0", "0", "0", "0", "0", "0", "3", "0", "0", "0", "0", "0", "0", "0", "0", "1", "0", "2", "0", "0", "0", "0", "4", "0", "0", "0", "1", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "1", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "5", "0", "0", "0", "0", "0", "0", "0", "2", "0", "0", "0", "0", "0", "0", "0", "0", "3", "0", "0", "0", "0", "0", "0" ]

Multiplicative with a(p^e) = e - 1.

A361431

[ "1", "2", "24", "34802", "509145568", "142743029326162", "715761543475698773496", "63014651062141097287201438690", "96683719664587866428237173383906926464", "2573179910450886540215919614478751310457090316706", "1184101051443285881265166362742300236491599013268534224381864" ]

Number of ways to write n^2 as an ordered sum of n^2 squares of integers.

A361432

[ "1", "1", "0", "1", "1", "0", "1", "2", "2", "0", "1", "3", "6", "4", "0", "1", "4", "12", "20", "8", "0", "1", "5", "20", "54", "68", "16", "0", "1", "6", "30", "112", "252", "232", "32", "0", "1", "7", "42", "200", "656", "1188", "792", "64", "0", "1", "8", "56", "324", "1400", "3904", "5616", "2704", "128", "0", "1", "9", "72", "490", "2628", "10000", "23360", "26568", "9232", "256", "0" ]

Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..floor(n/2)} k^(n-j) * binomial(n,2*j).

A361433

[ "1", "0", "1", "1", "0", "1", "1", "1", "0", "1", "1", "0", "1", "1", "0", "1", "1", "1", "0", "1", "1", "0", "1", "1", "1", "0", "1", "1", "0", "1", "1", "0", "1", "1", "1", "0", "1", "1", "0", "1", "1", "1", "0", "1", "1", "0", "1", "1", "1", "0", "1", "1", "0", "1", "1", "0", "1", "1", "1", "0", "1", "1", "0", "1", "1", "1", "0", "1", "1", "0", "1", "1", "0", "1", "1", "1", "0", "1", "1", "0", "1", "1", "1", "0", "1", "1" ]

a(n) = number of squares in the n-th antidiagonal of the natural number array, A000027.

A361434

[ "1", "4", "11", "18", "59", "108", "187", "198", "274", "335", "338", "374", "381", "387", "433", "815", "848", "1495", "1629", "2002", "3554", "3565", "4112", "4318", "4569", "4592", "4613", "4618", "4643", "4727", "4733", "6103", "6118", "7074", "7153", "7319", "7521", "7562", "7567", "7684", "7748", "7757", "7764", "7989", "8205", "8561", "8620" ]

Positions in Pi where the leader in the race of digits changes.

A361435

[ "1", "3", "11", "34", "144", "165", "229", "517", "790", "6870", "12757", "21134", "54155", "226470", "193225", "431900", "948949", "3960994", "6674779", "7594013", "14204939", "32720909", "20369309", "176923605", "335119938" ]

a(n) is the least positive integer that can be expressed as the sum of one or more consecutive squarefree numbers in exactly n ways.

A361437

[ "2", "3", "4", "5", "6", "7", "8", "12", "15", "58", "59", "102", "111", "118", "164", "291", "589", "685", "1671", "1900", "1945", "4905" ]

Numbers k such that k! - Sum_{i=1..k-1} (-1)^(k-i)*i! is prime.

A361438

[ "1", "1", "3", "1", "7", "1", "3", "5", "15", "1", "31", "1", "3", "7", "9", "21", "63", "1", "127", "1", "3", "5", "15", "17", "51", "85", "255", "1", "7", "73", "511", "1", "3", "11", "31", "33", "93", "341", "1023", "1", "23", "89", "2047", "1", "3", "5", "7", "9", "13", "15", "21", "35", "39", "45", "63", "65", "91", "105", "117", "195", "273", "315", "455", "585", "819", "1365", "4095", "1", "8191", "1", "3", "43", "127", "129", "381", "5461", "16383" ]

Triangle T(n,k), n >= 1, 1 <= k <= A046801(n), read by rows, where T(n,k) is k-th smallest divisor of 2^n-1.

A361442

[ "0", "1", "-1", "2", "-3", "4", "3", "-5", "8", "-12", "5", "-8", "13", "-21", "33", "6", "-11", "19", "-32", "53", "-86", "-2", "-4", "15", "-34", "66", "-119", "205", "9", "-7", "11", "-26", "60", "-126", "245", "-450", "10", "-19", "26", "-37", "63", "-123", "249", "-494", "944", "7", "-17", "36", "-62", "99", "-162", "285", "-534", "1028", "-1972" ]

Infinite triangle T(n, k), n, k >= 0, read and filled by rows the greedy way with distinct integers such that for any n, k >= 0, T(n, k) + T(n+1, k) + T(n+1, k+1) = 0; each term is minimal in absolute value and in case of a tie, preference is given to the positive value.

A361443

[ "0", "1", "2", "3", "5", "6", "-2", "9", "10", "7", "16", "-10", "24", "14", "17", "22", "-13", "-29", "-16", "-18", "-25", "-24", "-20", "-27", "-35", "12", "-30", "-42", "-22", "-36", "-40", "-43", "-44", "-45", "-46", "21", "35", "28", "32", "38", "27", "37", "41", "30", "50", "46", "55", "51", "56", "39", "74", "54", "73", "67", "57", "78", "71", "59", "61", "80", "68", "79" ]

a(n) is the first term of the n-th row of A361442.

A361444

[ "1", "2", "3", "4", "7", "6", "5", "8", "9", "22", "141", "88", "111", "202", "55", "222", "11", "212", "99", "232", "121", "252", "101", "66", "131", "242", "191", "272", "77", "282", "151", "292", "171", "262", "181", "606", "313", "414", "343", "444", "353", "44", "303", "424", "33", "434", "323", "404", "383", "474", "535", "484", "373", "454", "333", "464", "363" ]

Lexicographically earliest sequence of distinct positive base-10 palindromes such that a(n) + a(n+1) is prime.

A361445

[ "3", "5", "7", "11", "13", "11", "13", "17", "31", "163", "229", "199", "313", "257", "277", "233", "223", "311", "331", "353", "373", "353", "167", "197", "373", "433", "463", "349", "359", "433", "443", "463", "433", "443", "787", "919", "727", "757", "787", "797", "397", "347", "727", "457", "467", "757", "727", "787", "857", "1009", "1019", "857" ]

Sums of consecutive terms of A361444.

A361446

[ "1", "3", "16", "99", "717", "5964", "56701", "611750", "7432491", "100838222", "1514749135", "24989362186", "449429188211", "8754181791029", "183621843677724", "4126714250580949", "98932328702693666", "2520187379996442269", "67980528958530199837", "1935753445850303203221", "58025998739501873764826" ]

Number of connected 3-regular multigraphs on 2n unlabeled nodes rooted at an oriented edge (or loop), loops allowed.

A361447

[ "1", "2", "9", "49", "338", "2744", "26025", "282419", "3463502", "47439030", "718618117", "11937743088", "215896959624", "4224096594516", "88919920910684", "2004237153640098", "48165411560792500", "1229462431057436457", "33221743136066636436", "947415638925100675208", "28436953641282225835143" ]

Number of connected 3-regular (cubic) multigraphs on 2n unlabeled nodes rooted at an unoriented edge (or loop) whose removal does not disconnect the graph, loops allowed.

A361448

[ "1", "2", "10", "66", "511", "4536", "45519", "512661", "6436571", "89505875", "1369509795", "22908806774", "416408493351", "8178599551905", "172690849144538", "3902128758180500", "93970611848528998", "2402929936231885063", "65029668312580777779", "1856984518220396165657", "55803367549204703645086" ]

Number of connected 3-regular multigraphs on 2n unlabeled nodes rooted at an oriented edge (or loop) whose removal does not disconnect the graph, loops allowed.

A361449

[ "1", "4", "1573", "235862938", "37155328943771767", "12458003910177278332403197817", "15868284521418341362691384074620547198698934", "126024243590219798408446284849897811759970155660106999854057796", "9633603531065043175094488158875624821526224424118142906010095879389674957042528276201" ]

Number of colorings of an n X n grid up to permutation of the colors with no element having the same color as any horizontal, diagonal or antidiagonal neighbor.

A361450

[ "1", "5", "2906", "656404264", "148049849095504726", "67939294184937980415465539016", "114130286115375064054502412158789812265958284", "1159829070306179232444894822978404171908276758235252386883985596", "110658909677185498376669680234621983460781735371211477687464832774947897935655888426146" ]

Number of colorings of an n X n grid up to permutation of the colors with no element having the same color as any horizontal or antidiagonal neighbor.

A361451

[ "1", "2", "716", "112073062", "18633407199331522", "6575857942770612176290018153", "8769438200005128572266011359369913757287151", "72530091349507692706447958441062812294511923156598114466468667", "5746371835090565784276352813398004749296101606959968049467898643632416711373273639694" ]

Number of colorings of an n X n grid up to permutation of the colors with no element having the same color as any horizontal, vertical or antidiagonal neighbor.

A361452

[ "1", "7", "4192", "953124784", "213291369981652792", "96638817185266245591837984336", "160065721141038888919235753368205172658011648", "1603869086916486859475402575499346988054543498175515730927380336", "150972529586126094166343144224892296826763766718771806614594599643773846828229334720096" ]

Number of colorings of an n X n grid up to permutation of the colors with no element having the same color as any diagonal or antidiagonal neighbor.

A361453

[ "1", "15", "4141", "450288795", "50602429743064097", "12123635532529660182357354372" ]

Number of colorings of the n X n knight graph up to permutation of the colors.

A361454

[ "1", "4", "17", "78", "360", "1835", "10168", "62271", "419701", "3107800", "25108419", "219982357", "2076785950", "21011123423", "226708386212", "2598075587529", "31509529248585", "403155101535686", "5426659537490872", "76655160760249052", "1133766220709242638", "17522418780011531368", "282452568669871514771", "4740645804610572971112" ]

Number of 4-regular multigraphs on n unlabeled nodes with 4 external legs, loops allowed.

A361455

[ "1", "0", "1", "0", "1", "3", "0", "18", "21", "25", "0", "1606", "1173", "774", "543", "0", "565080", "271790", "122595", "59830", "29281", "0", "734774776", "229224750", "70500705", "25349355", "10110735", "3781503", "0", "3523091615568", "685793359804", "138122171880", "35130437825", "11002159455", "3767987307", "1138779265" ]

Triangle read by rows: T(n,k) is the number of simple digraphs on labeled n nodes with k strongly connected components.

A361456

[ "1", "1", "3", "2", "13", "30", "24", "6", "75", "372", "780", "872", "546", "180", "24", "541", "4660", "18180", "42140", "64150", "66900", "48320", "23820", "7650", "1440", "120", "4683", "62130", "385980", "1487520", "3973770", "7789032", "11565360", "13238520", "11771130", "8124710", "4314420", "1729440", "506010", "101880", "12600", "720" ]

Irregular triangle read by rows. T(n,k) is the number of properly colored simple labeled graphs on [n] with exactly k edges, n >= 0, 0 <= k <= binomial(n,2).

A361457

[ "3", "4", "6", "7", "8", "10", "11", "12", "14", "15", "16", "17", "19", "20", "21", "23", "24", "26", "27", "28", "29", "30", "33", "34", "35", "36", "37", "38", "40" ]

Numbers k such that the first player has a winning strategy in the game described in the Comments.

A361460

[ "0", "1", "0", "0", "1", "0", "1", "1", "0", "0", "1", "0", "1", "0", "1", "0", "1", "0", "1", "1", "0", "0", "1", "0", "0", "1", "1", "0", "1", "0", "1", "0", "0", "1", "1", "0", "1", "0", "1", "0", "1", "0", "1", "1", "0", "0", "1", "1", "1", "0", "0", "0", "1", "1", "1", "1", "0", "0", "1", "0", "1", "1", "1", "0", "1", "0", "1", "0", "1", "0", "1", "0", "1", "1", "1", "1", "0", "0", "1", "1", "0", "0", "1", "1", "0", "0", "1", "0", "1", "0", "0", "1", "0", "1", "1", "0", "1", "1", "1", "0", "1", "0", "1", "1", "0", "0", "1", "0", "1", "1", "1", "0", "1", "0", "0", "1", "0", "1", "1", "1" ]

a(n) = 1 if A135504(n+1) = 2 * A135504(n), otherwise 0.

A361461

[ "2", "5", "7", "8", "11", "13", "15", "17", "19", "20", "23", "26", "27", "29", "31", "34", "35", "37", "39", "41", "43", "44", "47", "48", "49", "53", "54", "55", "56", "59", "61", "62", "63", "65", "67", "69", "71", "73", "74", "75", "76", "79", "80", "83", "84", "87", "89", "92", "94", "95", "97", "98", "99", "101", "103", "104", "107", "109", "110", "111", "113", "116", "118", "119", "120", "123", "124", "125", "127", "129", "131", "132" ]

Numbers k such that x(k+1) = 2 * x(k), when x(1)=1 and x(n) = x(n-1) + lcm(x(n-1),n), i.e., x(n) = A135504(n).

A361462

[ "2", "1", "2", "1", "1", "3", "1", "1", "1", "3", "1", "1", "1", "1", "1", "1", "1", "3", "1", "1", "3", "3", "1", "1", "1", "1", "1", "1", "1", "3", "1", "3", "1", "1", "1", "1", "1", "1", "1", "1", "1", "3", "1", "1", "3", "3", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "3", "1", "1", "1", "1", "1", "1", "1", "3", "1", "3", "1", "3", "1", "1", "1", "1", "1", "1", "1", "3", "1", "1", "1", "3", "1", "1", "3", "1", "1", "1", "1", "1", "3", "1", "3", "1", "1", "1", "1", "1", "1", "1", "1", "3", "1", "1", "1", "3", "1", "1", "1", "1", "1", "1", "1", "3", "1", "1", "3", "1", "1", "1" ]

a(n) = A135506(n) mod 4.

A361463

[ "0", "0", "0", "0", "0", "1", "0", "0", "0", "1", "0", "0", "0", "0", "0", "0", "0", "1", "0", "0", "1", "1", "0", "0", "0", "0", "0", "0", "0", "1", "0", "1", "0", "0", "0", "0", "0", "0", "0", "0", "0", "1", "0", "0", "1", "1", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "1", "0", "0", "0", "0", "0", "0", "0", "1", "0", "1", "0", "1", "0", "0", "0", "0", "0", "0", "0", "1", "0", "0", "0", "1", "0", "0", "1", "0", "0", "0", "0", "0", "1", "0", "1", "0", "0", "0", "0", "0", "0", "0", "0", "1", "0", "0", "0", "1", "0", "0", "0", "0", "0", "0", "0", "1", "0" ]

a(n) = 1 if A135506(n) == 3 (mod 4), otherwise 0.

A361464

[ "6", "10", "18", "21", "22", "30", "32", "42", "45", "46", "58", "66", "68", "70", "78", "82", "85", "91", "93", "102", "106", "114", "117", "126", "128", "130", "133", "138", "140", "141", "150", "162", "165", "166", "171", "176", "178", "187", "190", "198", "200", "205", "210", "212", "213", "214", "222", "226", "234", "235", "238", "248", "250", "253", "261", "262", "267", "270", "282", "294", "301", "306", "308", "310", "320" ]

Numbers k such that A135504(k+1) / A135504(k) is a multiple of 4.

A361465

[ "1", "0", "1", "0", "0", "1", "1", "0", "0", "0", "0", "0", "0", "0", "1", "0", "0", "0", "0", "0", "1", "0", "0", "0", "0", "0", "0", "1", "0", "0", "1", "0", "1", "0", "0", "0", "0", "0", "0", "0", "0", "1", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "1", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "1", "0", "0", "0", "0", "0", "0", "1", "0", "1", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "1" ]

a(n) = 1 if A017665(n) [the numerator of the sum of the reciprocals of the divisors of n] is a power of 2, otherwise 0.

A361466

[ "1", "1", "0", "0", "1", "1", "0", "0", "0", "1", "0", "0", "0", "1", "0", "0", "0", "0", "0", "0", "0", "0", "0", "1", "0", "0", "0", "0", "1", "1", "0", "0", "0", "0", "0", "0", "0", "1", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "1", "0", "0", "1", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "1", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "1", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "1", "0", "0", "0", "0", "0", "0", "0", "1", "0", "0", "1", "0", "0", "0", "0", "0", "0", "1" ]

a(n) = 1 if A017665(A003961(n)) is a power of 2, otherwise 0. Here A017665 is the numerator of the sum of the reciprocals of the divisors of n, and A003961 is fully multiplicative with a(p) = nextprime(p).

A361467

[ "1", "12", "30", "117", "56", "360", "132", "1080", "775", "672", "182", "3510", "306", "1584", "1680", "9801", "380", "9300", "552", "6552", "3960", "2184", "870", "32400", "2793", "3672", "19500", "15444", "992", "20160", "1406", "88452", "5460", "4560", "7392", "90675", "1722", "6624", "9180", "60480", "1892", "47520", "2256", "21294", "43400", "10440", "2862", "294030", "16093", "33516", "11400" ]

a(n) = A003961(n) * sigma(A003961(n)), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.