christopher/oeis · Datasets at Hugging Face

a-number stringlengths 7 7	sequence sequencelengths 1 377	description stringlengths 3 852
A361468	[ "1", "12", "30", "117", "56", "40", "132", "1080", "775", "672", "182", "390", "306", "176", "1680", "9801", "380", "9300", "552", "6552", "3960", "2184", "870", "144", "2793", "408", "19500", "1716", "992", "2240", "1406", "88452", "5460", "4560", "7392", "90675", "1722", "736", "9180", "60480", "1892", "5280", "2256", "126", "43400", "1160", "2862", "32670", "16093", "3724", "456", "442", "3540", "26000" ]	a(n) = A249670(A003961(n)).
A361469	[ "0", "3", "3", "3", "4", "4", "4", "7", "3", "7", "3", "4", "4", "5", "7", "6", "4", "6", "5", "7", "7", "6", "4", "6", "4", "5", "7", "5", "6", "8", "3", "9", "6", "7", "8", "6", "4", "6", "7", "11", "4", "8", "6", "4", "7", "5", "5", "7", "4", "5", "5", "3", "5", "8", "5", "9", "8", "9", "3", "8", "4", "6", "7", "7", "8", "7", "6", "7", "5", "9", "3", "8", "6", "5", "7", "6", "7", "8", "5", "10", "6", "7", "5", "6", "8", "7", "9", "10", "4", "10", "8", "5", "6", "6", "9", "10", "4", "7", "6", "5", "5", "6", "6", "7", "11" ]	a(n) = bigomega(A249670(A003961(n))).
A361473	[ "1", "10", "27", "45", "143", "306", "903", "465", "1215", "3037", "2418", "4809", "17193", "8349", "32055", "75847", "117705", "306075", "379395", "467955", "1269075", "2517687", "1809295", "4720023", "6375915", "12961575", "21540987", "35647010", "16615305", "192717405", "268822806", "186269391", "247067415" ]	a(n) is the least positive integer that can be expressed as the sum of one or more consecutive nonprime numbers in exactly n ways.
A361475	[ "0", "1", "0", "3", "1", "0", "7", "4", "1", "0", "15", "13", "5", "1", "0", "31", "40", "21", "6", "1", "0", "63", "121", "85", "31", "7", "1", "0", "127", "364", "341", "156", "43", "8", "1", "0", "255", "1093", "1365", "781", "259", "57", "9", "1", "0", "511", "3280", "5461", "3906", "1555", "400", "73", "10", "1", "0", "1023", "9841", "21845", "19531", "9331", "2801", "585", "91", "11", "1", "0" ]	Array read by ascending antidiagonals: A(n, k) = (k^n - 1)/(k - 1), with k >= 2.
A361476	[ "0", "1", "4", "12", "34", "99", "308", "1040", "3820", "15197", "65060", "297828", "1449742", "7468527", "40555732", "231335944", "1381989864", "8623700793", "56078446596", "379233142780", "2662013133274", "19362917621979", "145719550012276", "1133023004941248", "9090156910550084", "75161929739797493", "639793220877941476" ]	Antidiagonal sums of A361475.
A361477	[ "1", "1", "1", "1", "2", "1", "2", "1", "2", "3", "1", "3", "1", "3", "2", "1", "2", "3", "4", "3", "4", "1", "4", "3", "2", "3", "4", "3", "2", "3", "2", "1", "2", "3", "4", "6", "6", "5", "6", "6", "4", "5", "1", "5", "6", "5", "4", "3", "2", "6", "6", "1", "6", "5", "6", "6", "1", "6", "4", "6", "2", "3", "2", "1", "2", "3", "4", "6", "12", "5", "12", "3", "12", "10", "6", "10", "4", "10", "12", "6", "4", "5", "6", "10" ]	a(n) is the number of integers whose binary expansions have the same multiset of run-lengths as that of n.
A361478	[ "0", "1", "2", "3", "4", "6", "5", "4", "6", "7", "8", "14", "9", "11", "13", "10", "9", "11", "13", "12", "9", "11", "13", "8", "14", "15", "16", "30", "17", "23", "29", "18", "20", "22", "26", "19", "25", "27", "18", "20", "22", "26", "21", "18", "20", "22", "26", "17", "23", "29", "24", "28", "19", "25", "27", "18", "20", "22", "26", "19", "25", "27", "24", "28", "17", "23", "29", "16", "30" ]	Irregular table T(n, k), n >= 0, k = 1..A361477(n), read by rows; the n-th row lists the integers whose binary expansions have the same multiset of run-lengths as that of n.
A361479	[ "0", "1", "2", "3", "4", "5", "4", "7", "8", "9", "10", "9", "12", "9", "8", "15", "16", "17", "18", "19", "18", "21", "18", "17", "24", "19", "18", "19", "24", "17", "16", "31", "32", "33", "34", "35", "36", "37", "36", "35", "34", "37", "42", "37", "36", "37", "34", "33", "48", "35", "36", "51", "36", "37", "36", "35", "56", "35", "34", "35", "48", "33", "32", "63", "64", "65", "66", "67" ]	a(n) is the least integer whose binary expansion has the same multiset of run-lengths as that of n.
A361480	[ "0", "1", "2", "3", "6", "5", "6", "7", "14", "13", "10", "13", "12", "13", "14", "15", "30", "29", "26", "27", "26", "21", "26", "29", "28", "27", "26", "27", "28", "29", "30", "31", "62", "61", "58", "59", "54", "53", "54", "59", "58", "53", "42", "53", "54", "53", "58", "61", "60", "59", "54", "51", "54", "53", "54", "59", "56", "59", "58", "59", "60", "61", "62", "63", "126", "125" ]	a(n) is the greatest integer whose binary expansion has the same multiset of run-lengths as that of n.
A361481	[ "0", "1", "2", "3", "4", "6", "5", "7", "8", "14", "9", "11", "13", "10", "12", "15", "16", "30", "17", "23", "29", "18", "20", "22", "26", "19", "25", "27", "21", "24", "28", "31", "32", "62", "33", "47", "61", "34", "40", "46", "58", "35", "39", "49", "55", "57", "59", "36", "38", "44", "50", "52", "54", "37", "41", "43", "45", "53", "42", "48", "60", "51", "56", "63", "64", "126", "65" ]	Distinct values of A361478, in order of appearance.
A361482	[ "0", "1", "2", "3", "4", "6", "5", "7", "8", "10", "13", "11", "14", "12", "9", "15", "16", "18", "21", "25", "22", "28", "23", "19", "29", "26", "24", "27", "30", "20", "17", "31", "32", "34", "37", "41", "47", "53", "48", "42", "38", "54", "58", "55", "49", "56", "39", "35", "59", "43", "50", "61", "51", "57", "52", "44", "62", "45", "40", "46", "60", "36", "33", "63", "64", "66", "69", "73" ]	Inverse permutation to A361481.
A361483	[ "7", "13", "37", "61", "97", "103", "127", "163", "193", "211", "223", "307", "313", "331", "337", "397", "421", "463", "487", "541", "571", "601", "607", "631", "673", "691", "727", "757", "853", "907", "937", "967", "1021", "1033", "1051", "1063", "1117", "1153", "1171", "1231", "1237", "1297", "1303", "1327", "1381", "1453", "1531", "1567", "1621", "1657", "1693", "1723" ]	Primes p such that p + 256 is also prime.
A361484	[ "11", "29", "59", "89", "101", "107", "131", "149", "179", "197", "227", "239", "257", "311", "317", "347", "479", "509", "521", "557", "617", "641", "659", "701", "719", "809", "887", "911", "941", "947", "971", "977", "1019", "1031", "1097", "1109", "1151", "1181", "1187", "1229", "1277", "1289", "1319", "1361", "1367", "1439", "1481", "1487", "1499", "1571", "1601" ]	Primes p such that p + 512 is also prime.
A361485	[ "7", "37", "67", "73", "79", "127", "139", "157", "163", "193", "199", "277", "283", "337", "349", "409", "457", "463", "487", "499", "547", "577", "613", "643", "673", "709", "787", "823", "853", "877", "883", "907", "1039", "1063", "1087", "1117", "1129", "1213", "1249", "1327", "1399", "1423", "1453", "1567", "1597", "1609", "1663", "1669", "1753", "1777", "1873", "1879" ]	Primes p such that p + 1024 is also prime.
A361486	[ "1", "1", "1", "1", "2", "2", "3", "2", "2", "3", "2", "2", "3", "2", "2", "3", "1", "3", "3", "1", "4", "1", "4", "3", "5", "5", "1", "4", "3", "4", "5", "4", "4", "5", "6", "6", "7", "4", "4", "5", "5", "6", "2", "4", "1", "4", "5", "1", "6", "2", "6", "4", "6", "5", "5", "7", "2", "3", "4", "6", "5", "5", "7", "2", "3", "8", "1", "4", "3", "6", "7", "5", "5", "3", "5", "7", "6", "3", "1", "1", "7", "8", "7", "7", "4", "5", "8", "5", "9", "6", "6", "8", "7", "7", "6", "8", "9", "9", "3" ]	Lexicographically earliest sequence of positive numbers on a square spiral such that no three equal numbers are collinear.
A361488	[ "1", "0", "0", "2", "2", "0", "6", "12", "6", "20", "60", "60", "90", "280", "420", "532", "1330", "2520", "3444", "6804", "14112", "21912", "37884", "77616", "133914", "223080", "432432", "793364", "1341912", "2471040", "4629196", "8076640", "14453010", "26960232", "48308832", "85794852", "157947816", "287413152", "512697900", "933072064" ]	Diagonal of rational function 1/(1 - (x^3 + y^3 + x^4*y)).
A361489	[ "1", "1", "2", "6", "19", "72", "313", "1472", "7612", "42679", "255515", "1632710", "11065057", "79065807", "594174922", "4679473130", "38500353667", "330172915164", "2944613004359", "27253908250340", "261328607398332", "2591724561444621", "26545170005412613", "280411070646125638" ]	Expansion of e.g.f. exp(exp(x) - 1 + x^3/6).
A361490	[ "8", "45", "52", "75", "92", "99", "130", "147", "164", "195", "236", "255", "266", "333", "406", "423", "430", "477", "494", "555", "574", "627", "670", "711", "716", "777", "782", "801", "806", "903", "908", "915", "932", "935", "938", "969", "1010", "1017", "1022", "1065", "1076", "1233", "1244", "1443", "1474", "1479", "1490", "1533", "1556", "1635", "1724", "1737", "1790", "1833", "1844", "2007", "2012" ]	a(1) = 8; for n > 1, a(n) is the least triprime > a(n-1) such that a(n) - a(n-1) and a(n) + a(n-1) are both prime.
A361491	[ "1", "73", "2521", "85681", "2910673", "98877241", "3358915561", "114104251873", "3876185648161", "131676207785641", "4473114879063673", "151954229680379281", "5161970694253831921", "175355049374949906073", "5956909708054042974601", "202359575024462511230401", "6874268641123671338859073", "233522774223180363009978121" ]	Expansion of x(1+38x+x^2)/((1-x)(x^2-34x+1)).
A361492	[ "1", "1", "2", "6", "30", "30", "210", "210", "210", "17430", "30030", "60060", "510510", "3573570" ]	Common difference corresponding to increasing arithmetic progression of at least n >= 2 primes whose first term is A284708(n); a(1) = 1.
A361493	[ "1", "1", "2", "11", "39", "172", "1163", "6547", "41772", "335139", "2486215", "20078610", "186139957", "1676540257", "16077206122", "168739976555", "1763716943267", "19358116589964", "226362412711759", "2669223655597955", "32748447519013132", "421204995451111971", "5496921281576148363" ]	Expansion of e.g.f. exp(exp(x) - 1 + x^3).
A361497	[ "3", "4", "4", "4", "7", "7", "6", "8", "6", "10", "12", "6", "11", "8", "12", "14", "10", "17", "14", "16", "16", "16", "15", "8", "20", "21", "10", "18", "14", "24", "18", "20", "12", "16", "22", "25", "30", "20", "22", "12", "32", "36", "12", "30", "16", "42", "22", "32", "16", "22", "30", "31", "32", "22", "26", "12", "36", "39", "18", "37", "28", "48", "35", "28", "32", "42", "32", "34", "46", "30", "29", "26", "40", "49", "52", "14", "45", "24", "34", "48", "32", "56", "30", "44", "38", "40", "36", "45", "20", "54", "74", "22", "49", "28" ]	Number of cusps in Prym-Teichmuller curve W_D(4) of discriminant D = A361169(n).
A361498	[ "0", "0", "0", "0", "0", "0", "0", "0", "0", "1", "1", "1", "1", "2", "1", "1", "2", "1", "1", "3", "5", "1", "4", "4", "6", "4", "5", "7", "5", "9", "6", "9", "8", "9", "10", "9", "8", "9", "9", "9", "11", "14", "10", "11", "14", "15", "16", "14", "15", "14", "21", "18", "22", "19", "19", "17", "23", "21", "17", "23", "19", "27", "22", "30", "18", "27", "22", "34", "30", "25", "29", "22", "41", "35", "43", "26", "36", "29", "33", "36", "30", "39", "35", "39", "43", "55", "37", "40", "38", "49", "46", "38", "44", "40" ]	Genus of Prym-Teichmuller curve W_D(4) of discriminant D = A361169(n).
A361499	[ "0", "1", "1", "0", "0", "0", "1", "0", "1", "0", "1", "2", "0", "0", "0", "2", "1", "0", "1", "0", "2", "1", "0", "0", "0", "0", "3", "0", "1", "0", "0", "3", "2", "0", "0", "0", "2", "2", "0", "2", "0", "1", "3", "0", "0", "0", "0", "4", "2", "1", "0", "0", "2", "2", "0", "0", "0", "0", "3", "0", "2", "0", "0", "3", "3", "0", "0", "0", "2", "1", "0", "3", "0", "0", "3", "4", "0", "0", "0", "4", "4", "0", "1", "0", "0", "4", "2", "0", "0", "0", "2", "5", "0", "2" ]	Number of orbifold points of order 2 in Prym-Teichmuller curve W_D(4) of discriminant D = A361169(n).
A361500	[ "1", "0", "0", "2", "0", "0", "2", "1", "2", "0", "0", "2", "1", "0", "2", "0", "0", "2", "2", "0", "0", "4", "1", "4", "0", "2", "2", "0", "0", "0", "3", "0", "2", "2", "0", "1", "0", "2", "3", "4", "2", "0", "4", "0", "2", "0", "2", "0", "0", "6", "0", "0", "0", "4", "2", "4", "0", "4", "4", "1", "0", "0", "2", "0", "0", "4", "4", "0", "0", "6", "5", "2", "0", "2", "0", "6", "0", "0", "3", "0", "2", "2", "6", "0", "2", "0", "4", "3", "4", "0", "0", "6", "0", "2" ]	Number of orbifold points of order 3 in Prym-Teichmuller curve W_D(4) of discriminant D = A361169(n).
A361501	[ "11", "112", "1124", "11248", "1124816", "112481623", "11248162328", "1124816232838", "112481623283849", "11248162328384962", "1124816232838496270", "112481623283849627077", "2486232838496270779", "248623283849627077997", "248623283849627077997113", "248623283849627077997113118", "248623283849627077997113118128" ]	A variant of A359143 in which all copies of a digit d are erased only when d is both the leading digit and the final digit of (a(n) concatenated with sum of digits of a(n)).
A361502	[ "2", "3", "4", "8", "13", "42", "347", "3466", "49012", "528231", "717126", "63056215", "1375559400", "7038527851" ]	Index of n-th prime in A359804.
A361503	[ "2", "3", "5", "2", "3", "5", "7", "3", "2", "5", "7", "11", "3", "5", "7", "11", "5", "3", "7", "5", "11", "7", "2", "3", "5", "7", "11", "5", "7", "2", "3", "5", "7", "3", "5", "7", "2", "5", "7", "11", "13", "5", "2", "3", "5", "2", "7", "3", "5", "7", "11", "5", "3", "2", "5", "7", "11", "2", "5", "3", "7", "5", "3", "7", "2", "11", "3", "7", "5", "3", "7", "5", "11", "7", "5", "11", "7", "5", "3", "7", "11", "3", "2", "5", "7", "2", "5", "3", "2", "5", "7", "13", "3", "5", "2", "3" ]	a(1)=2; thereafter a(n) = smallest prime that does not divide b(n-1)*b(n), where b(k) = A359804(k).
A361504	[ "1", "2", "3", "5", "4", "6", "8", "10", "9", "7", "13", "14", "42", "12", "11", "24", "347", "19", "3466", "15", "16", "17", "49012", "25", "18", "31", "32", "20", "528231", "21", "717126", "38", "22", "44", "23", "35", "63056215", "47", "45", "26", "1375559400", "27", "7038527851", "28", "29", "55" ]	Index of n in A359804, or -1 if n never appears there.
A361505	[ "1", "2", "5", "10", "24", "38", "87", "172", "349", "706", "1407", "2752", "5487", "11103", "22285", "44429", "88993", "177746", "356460", "712129", "1425163", "2849424", "5701776", "11401709", "22804522", "45608572", "91219022", "182438457", "364879209", "729757797", "1459514883", "2919031155", "5838065175", "11676129412", "23352260426" ]	Index of 2^n in A359804.
A361506	[ "1", "3", "8", "19", "45", "102", "232", "524", "1181", "2659", "5984", "13466", "30300", "68177", "153400", "345151", "776590", "1747330", "3931495", "8845865", "19903197", "44782195", "100759940", "226709865", "510097199", "1147718700", "2582367075", "5810325920", "13073233320", "29414774970", "66183243690", "148912298300", "335052671200", "753868510200" ]	a(n) = floor( (4/5)*( (9/4)^(n+1)-1 ) ).
A361507	[ "1", "3", "7", "16", "37", "84", "190", "428", "964", "2170", "4883", "10987", "24721", "55623", "125152", "281593", "633585", "1425567", "3207526", "7216934", "16238102", "36535730", "82205393", "184962135", "416164804", "936370810", "2106834323", "4740377227", "10665848761", "23998159713", "53995859355", "121490683549", "273354037986", "615046585469", "1383854817306", "3113673338939" ]	a(0) = 1; thereafter a(n) = floor((9/4)*a(n-1)) + 1.
A361517	[ "3", "4", "5", "11", "17", "27", "35", "37", "49", "59", "69", "81", "91", "103", "115", "123", "135", "137", "167", "175", "189", "199", "207", "287", "295", "307", "361", "1051", "2507", "2757", "2917", "3057", "3081", "7255", "7361", "7871", "16173" ]	The value of n for which the two-player impartial {0,1}-Toggle game on a generalized Petersen graph GP(n,2) with a (1,0)-weight assignment is a next-player winning game.
A361521	[ "0", "0", "0", "0", "2", "0", "0", "5", "4", "0", "0", "9", "12", "6", "0", "0", "14", "24", "21", "8", "0", "0", "20", "40", "45", "32", "10", "0", "0", "27", "60", "78", "72", "45", "12", "0", "0", "35", "84", "120", "128", "105", "60", "14", "0", "0", "44", "112", "171", "200", "190", "144", "77", "16", "0", "0", "54", "144", "231", "288", "300", "264", "189", "96", "18", "0" ]	Array read by descending antidiagonals. A(n, k) is the number of the nonempty multiset combinations of {0, 1} as defined in A361682.
A361522	[ "1", "0", "1", "0", "2", "0", "6", "0", "24", "0", "120", "0", "720", "0", "5040", "0", "40320", "0", "362880", "0", "3628800", "0", "39916800", "0", "479001600", "0", "6227020800", "0", "87178291200", "0", "1307674368000", "0", "20922789888000", "0", "355687428096000", "0", "6402373705728000", "0", "121645100408832000", "0", "2432902008176640000" ]	The aerated factorial numbers.
A361523	[ "1", "1", "1", "1", "2", "4", "1", "3", "17", "54", "1", "6", "61", "892", "9235", "1", "10", "220", "8159", "406653", "10538496" ]	Triangle read by rows: T(n,k) is the number of ways of dividing an n X k rectangle into integer-sided rectangles, up to rotations and reflections.
A361524	[ "1", "1", "4", "54", "9235", "10538496" ]	Number of ways of dividing an n X n square into integer-sided rectangles, up to rotations and reflections.
A361525	[ "1", "3", "17", "54", "892", "8159", "80021", "791821", "7906439", "79069308" ]	Number of ways of dividing an n X 3 rectangle into integer-sided rectangles, up to rotations and reflections.
A361526	[ "1", "6", "61", "892", "9235", "406653", "9252097", "211703640" ]	Number of ways of dividing an n X 4 rectangle into integer-sided rectangles, up to rotations and reflections.
A361527	[ "1", "0", "1", "0", "1", "3", "0", "2", "21", "25", "0", "6", "213", "774", "543", "0", "24", "3470", "30275", "59830", "29281", "0", "120", "95982", "1847265", "7757355", "10110735", "3781503", "0", "720", "4578588", "190855000", "1522899105", "3944546095", "3767987307", "1138779265" ]	Triangular array read by rows. T(n,k) is the number of labeled digraphs on [n] having exactly k strongly connected components all of which are simple cycles, n >= 0, 0 <= k <= n.
A361528	[ "1", "1", "8", "75", "804", "9681", "129168", "1889379", "30037500", "515342817", "9484627608", "186305208219", "3888697965012", "85920579594225", "2002828537732896", "49107722192594739", "1263165207424720812", "34004577057249890241", "955970215914084949800", "28011115058953357075563", "853924857091970071203972" ]	a(n) = (2+n)(2a(n-1) - (n-2)*a(n-2)) with a(0)=a(1)=1.
A361530	[ "23", "37", "53", "73", "113", "127", "131", "137", "139", "151", "157", "173", "179", "193", "197", "211", "223", "229", "233", "239", "241", "271", "283", "293", "311", "313", "317", "331", "337", "347", "353", "359", "367", "373", "379", "383", "389", "397", "421", "431", "433", "457", "523", "541", "547", "571", "593", "613", "617", "631", "673", "677", "719" ]	Primes that can be written as the result of shuffling the decimal digits of two primes.
A361531	[ "1", "-1", "0", "2", "-3", "-2", "21", "-44", "-62", "631", "-1367", "-3170", "34849", "-86855", "-302964", "3058342", "-8509971", "-36488802", "430842051", "-1111575888", "-6244999438", "78663444549", "-250850311489", "-1724880111306", "18475299723737", "-65061274823853", "-444914618968648", "6831921081061986" ]	Expansion of e.g.f. exp(1 - exp(x) + x^3/6).
A361532	[ "1", "1", "4", "19", "118", "886", "7786", "78184", "881644", "11017108", "150966856", "2249261356", "36181351504", "624658612384", "11516406883528", "225740649754936", "4686671645814736", "102712289940757264", "2369128149877075264", "57359541280704038128", "1454229915957292684576" ]	Expansion of e.g.f. exp((x + x^2/2)/(1-x)).
A361533	[ "1", "0", "0", "1", "4", "20", "130", "980", "8400", "80920", "865200", "10164000", "130114600", "1802600800", "26867640800", "428661633400", "7288513232000", "131558835408000", "2512282795422400", "50600743739145600", "1071998968264224000", "23829055696093648000", "554524256514356128000" ]	Expansion of e.g.f. exp(x^3/(6 * (1-x))).
A361535	[ "1", "10", "61", "290", "1172", "4212", "13833", "42262", "121625", "332764", "871641", "2197936", "5359005", "12679730", "29200593", "65617892", "144189054", "310400110", "655669910", "1360910666", "2779007594", "5589070978", "11081585154", "21679798590", "41883282555", "79958881544", "150943109191", "281926365224" ]	Expansion of g.f. 1 / Product_{n>=1} ((1 - x^n)^6 * (1 - x^(2*n-1))^4).
A361536	[ "1", "3", "60", "2037", "92187", "5066952", "322801089", "23197971285", "1848188250810", "161297106209607", "15285968218925460", "1562519987561305566", "171348519312001997550", "20068058089211306151393", "2500498134501774994768119", "330350627790472265384885061", "46136067767500181432129130897" ]	Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(3n) A(x)^(3*n) / n!.
A361537	[ "1", "3", "75", "3234", "186471", "13063908", "1060481214", "97053553710", "9840717984447", "1092337371705273", "131589391554509112", "17089208887923714204", "2379797411747290723350", "353790840030976298935989", "55935780589531899802966062", "9373903063348266793396858620" ]	Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(3n) A(x)^(4*n) / n!.
A361538	[ "1", "5", "244", "19090", "1839075", "199363606", "23320604384", "2876887986028", "369107300988219", "48807370374419910", "6610055144592717246", "912769451975745598299", "128087096344387852459658", "18219369599509083643661044", "2621701165622760015979876800", "381039123615762485580377457688" ]	Central terms of triangle A361050.
A361539	[ "3", "39", "426", "4550", "50085", "577731", "7022596", "90148860", "1222753815", "17515226465", "264663151038", "4212100028994", "70475063838361", "1237144088015535", "22735980569119560", "436467520716475064", "8733235757816095083", "181740089309026259565", "3925458146197916823970" ]	a(n) = A361540(n, n-2) for n >= 2, a diagonal of triangle A361540.
A361540	[ "1", "1", "1", "3", "4", "1", "22", "39", "18", "1", "269", "604", "426", "92", "1", "4616", "12625", "12040", "4550", "520", "1", "102847", "332766", "401355", "218300", "50085", "3222", "1", "2824816", "10574725", "15456756", "11017895", "3867080", "577731", "21700", "1", "92355769", "393171416", "676130644", "597596216", "284455150", "69038984", "7022596", "157544", "1" ]	Expansion of e.g.f. A(x,y) satisfying A(x,y) = Sum_{n>=0} (A(x,y)^n + y)^n * x^n/n!, as a triangle read by rows.
A361541	[ "1", "4", "56", "1220", "34788", "1203152", "48418384", "2210163032", "112501779300", "6308565897088", "386149471644704", "25614932030415636", "1830512170952711968", "140224558208217547440", "11464991752291729651224", "996723500374559386157920", "91824970792933898453830680" ]	Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(4n) A(x)^n / n!.
A361542	[ "1", "4", "84", "2940", "137228", "7809680", "517517212", "38860889496", "3248881861500", "298704250964336", "29928006672383280", "3244628959712243628", "378449007991303855532", "47261928190105905687600", "6293239981401396941576632", "890249832854933140207681360", "133355904852469516343820132852" ]	Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(4n) A(x)^(2*n) / n!.
A361543	[ "1", "4", "112", "5380", "346788", "27285968", "2498963752", "259124694312", "29885849525700", "3786931724896768", "522451837498888672", "77929657518224116484", "12496899169394954817144", "2144326582901160246138160", "392104633203721656029928184", "76134826269461672101153285664" ]	Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(4n) A(x)^(3*n) / n!.
A361544	[ "1", "4", "39", "604", "12625", "332766", "10574725", "393171416", "16744363569", "803841993370", "42957812253301", "2529951235854516", "162852898603253209", "11378885054925777494", "858009440175419213445", "69471138931959493061296", "6013997809048628612191585", "554545575488282609142617778" ]	a(n) = A361540(n,1) for n >= 1, a column of triangle A361540.
A361545	[ "1", "0", "0", "0", "1", "5", "30", "210", "1715", "15750", "160650", "1801800", "22043175", "292116825", "4168464300", "63725161500", "1039028615625", "17998106626500", "330068683444500", "6388785205803000", "130156170633113625", "2783924007745505625", "62375052003905891250", "1460924768552182683750" ]	Expansion of e.g.f. exp(x^4/(24 * (1-x))).
A361546	[ "1", "5", "3", "5", "9", "5", "9", "11", "45", "23", "35", "15", "3", "9", "27", "51", "27", "53", "9", "39", "23", "249", "51", "51", "131", "221", "29", "105", "321", "179", "5", "221", "111", "411", "191", "65", "83", "75", "95", "101", "147", "83", "149", "111", "203", "131", "9", "245", "281", "15", "83", "65", "299", "39", "51", "51", "225", "65", "81", "125", "611", "143", "65", "107", "21" ]	a(n) is the least odd number k such that k*2^prime(n) + 1 is prime, or -1 if no such number k exists.
A361547	[ "1", "0", "0", "0", "0", "1", "6", "42", "336", "3024", "30366", "335412", "4041576", "52756704", "741620880", "11169844686", "179448036768", "3063069801792", "55360031126400", "1056123043335360", "21208345049147256", "447183762148547424", "9877939209960101280", "228112734232663600320" ]	Expansion of e.g.f. exp(x^5/(120 * (1-x))).
A361548	[ "1", "1", "4", "20", "126", "966", "8656", "88544", "1016380", "12920156", "179996816", "2725070096", "44521522024", "780344770440", "14599772973696", "290311643773376", "6112190642062096", "135798496839920144", "3174483084427144000", "77872118431269176896", "1999809157085214044896" ]	Expansion of e.g.f. exp((x + x^2/2 + x^3/6)/(1-x)).
A361549	[ "1", "18", "426", "12040", "401355", "15456756", "676130644", "33151425840", "1802216703285", "107652497473180", "7012494336544686", "494963689847333928", "37648456802884402111", "3071415347513049808740", "267644521958509484952360", "24822151072519637091258976", "2442314922307988498911793385" ]	a(n) = A361540(n,2) for n >= 2, a column of triangle A361540.
A361550	[ "1", "0", "1", "0", "5", "1", "0", "18", "10", "1", "0", "55", "61", "20", "1", "0", "149", "290", "215", "35", "1", "0", "371", "1172", "1660", "555", "56", "1", "0", "867", "4212", "10311", "5850", "1254", "84", "1", "0", "1923", "13833", "54688", "47460", "17773", "2555", "120", "1", "0", "4086", "42262", "256815", "319409", "188300", "46844", "4810", "165", "1", "0", "8374", "121625", "1093790", "1864445", "1621116", "621915", "111348", "8505", "220", "1", "0", "16634", "332764", "4297370", "9717550", "11913160", "6557572", "1818022", "243795", "14290", "286", "1" ]	Expansion of g.f. A(x,y) satisfying xy = Sum_{n=-oo..+oo} x^(n(3n+1)/2) (A(x,y)^(3n) - 1/A(x,y)^(3n+1)), as a triangle read by rows.
A361551	[ "1", "5", "90", "2535", "93840", "4226355", "222038775", "13259599965", "884588496165", "65114097133590", "5239173990133060", "457392343670390700", "43064135370809341250", "4350264113638902544555", "469422682906897831519170", "53897717818214315584719430", "6561919113715122121302125775" ]	Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n (x^(5n) A(x)^n) / n!.
A361552	[ "1", "2", "14", "84", "530", "3770", "29446", "240302", "2003914", "17024332", "147306448", "1294859540", "11524690228", "103605031978", "939357512086", "8580744729478", "78898896072996", "729661925134886", "6782435427053490", "63332055630823770", "593793935288453260", "5587934788557993846" ]	Expansion of g.f. A(x) satisfying 2x = Sum_{n=-oo..+oo} x^(n(3n+1)/2) (A(x)^(3n) - 1/A(x)^(3n+1)).
A361553	[ "1", "3", "24", "171", "1335", "11940", "115773", "1160901", "11901537", "124726644", "1332688035", "14455451526", "158660036535", "1758835084221", "19667067522966", "221573079684087", "2512635069594897", "28656903391830291", "328500210705228867", "3782806859877522522", "43738575934977450465" ]	Expansion of g.f. A(x) satisfying 3x = Sum_{n=-oo..+oo} x^(n(3n+1)/2) (A(x)^(3n) - 1/A(x)^(3n+1)).
A361554	[ "1", "4", "36", "296", "2732", "28980", "329996", "3872908", "46575260", "573472248", "7197096168", "91640952360", "1180636398320", "15364364313588", "201691201775092", "2667523242203932", "35510152549696208", "475424653523498396", "6397601663340197268", "86481499341290372804", "1173813146742741571560" ]	Expansion of g.f. A(x) satisfying 4x = Sum_{n=-oo..+oo} x^(n(3n+1)/2) (A(x)^(3n) - 1/A(x)^(3n+1)).
A361555	[ "1", "5", "50", "465", "4925", "59870", "776155", "10364135", "142082065", "1995371980", "28549274995", "414327073520", "6084353526535", "90258375062245", "1350607531232830", "20361436162127965", "308964002231172075", "4715119823819824535", "72324133311820587435", "1114404268419043050750" ]	Expansion of g.f. A(x) satisfying 5x = Sum_{n=-oo..+oo} x^(n(3n+1)/2) (A(x)^(3n) - 1/A(x)^(3n+1)).
A361556	[ "1", "5", "61", "1660", "47460", "1621116", "58002140", "2213389940", "87301563690", "3555890156445", "148125509781095", "6292884402884976", "271565202254735207", "11878392121526009800", "525519782174930309205", "23481280252471520720288", "1058270749214634093475910", "48058678036035725619136698" ]	Central terms of triangle A361550.
A361557	[ "1", "1", "4", "20", "127", "977", "8789", "90267", "1040260", "13275258", "185653535", "2821321725", "46265262553", "813871304989", "15281792484768", "304949014412540", "6442741397501699", "143633948442619765", "3369004776395733829", "82919378806522132407", "2136425765494805888952" ]	Expansion of e.g.f. exp((exp(x) - 1)/(1-x)).
A361558	[ "1", "1", "4", "20", "127", "976", "8776", "90084", "1037555", "13233077", "184956386", "2809098986", "46038214729", "809411443790", "15189361799522", "302932433571356", "6396529241755881", "142523960797017589", "3341115707515530400", "82187749261419720712", "2116421112495023612311" ]	Expansion of e.g.f. exp((x + x^2/2 + x^3/6 + x^4/24)/(1-x)).
A361559	[ "0", "10", "258", "1740", "20070", "48510", "196920", "350370", "937860", "3075030", "4322160", "10641330", "17925180", "22825110", "35827560", "65816010", "113180910", "133937670", "215070570", "288148140", "331474860", "493573080", "633015810", "899599140", "1387338960", "1700082450", "1876303260", "2272556790", "2494333710" ]	a(n) = Sum_{k=1..prime(n)-1} floor(k^5/prime(n)).
A361560	[ "1", "1", "4", "47", "1471", "115042", "21591817", "9455689609", "9464951556046", "21316993121024757", "106689322228222150243", "1174731578884501228621956", "28221161668500867009724237123", "1468937207982284446757761131062629", "164682046577167683717133576752582349216", "39562388056404531283767850863430043742371123" ]	Number of labeled digraphs on [n] all of whose strongly connected components are complete digraphs.
A361562	[ "3", "7", "11", "19", "23", "31", "43", "79", "127", "167", "191", "199", "347", "3539", "5807", "10691", "11279", "12391", "14479", "83339", "117239", "127031", "141079", "269987", "986191", "4031399" ]	Wagstaff numbers that are of the form 4*k + 3.
A361563	[ "5", "13", "17", "61", "101", "313", "701", "1709", "2617", "10501", "42737", "95369", "138937", "267017", "374321" ]	Wagstaff numbers that are of the form 4*k + 1.
A361564	[ "4", "6", "10", "17", "25", "39", "59", "87", "127", "186" ]	Number of (n-3)-connected unlabeled n-node graphs.
A361565	[ "1", "3", "2", "2", "3", "5", "4", "3", "3", "7", "6", "7", "7", "9", "4", "4", "9", "9", "10", "9", "5", "13", "12", "5", "5", "15", "6", "11", "15", "11", "16", "6", "7", "19", "6", "6", "19", "21", "8", "13", "21", "13", "22", "15", "7", "25", "24", "7", "7", "15", "10", "17", "27", "15", "8", "15", "11", "31", "30", "8", "31", "33", "8", "8", "9", "17", "34", "21", "13", "17", "36", "17", "37", "39", "10" ]	a(n) is the numerator of the median of divisors of n.
A361566	[ "1", "2", "1", "1", "1", "2", "1", "1", "1", "2", "1", "2", "1", "2", "1", "1", "1", "2", "1", "2", "1", "2", "1", "1", "1", "2", "1", "2", "1", "2", "1", "1", "1", "2", "1", "1", "1", "2", "1", "2", "1", "2", "1", "2", "1", "2", "1", "1", "1", "2", "1", "2", "1", "2", "1", "2", "1", "2", "1", "1", "1", "2", "1", "1", "1", "2", "1", "2", "1", "2", "1", "2", "1", "2", "1", "2", "1", "2", "1", "1", "1", "2", "1", "2", "1", "2", "1", "2" ]	a(n) is the denominator of the median of divisors of n.
A361567	[ "1", "0", "1", "6", "15", "60", "555", "3150", "17745", "158760", "1399545", "10914750", "102920895", "1104323220", "11249313075", "119330961750", "1426411411425", "17429852840400", "213417453474225", "2791671804271350", "38524272522310575", "537569719902715500", "7732658753799054075" ]	Expansion of e.g.f. exp(x^2/2 * (1+x)^2).
A361568	[ "1", "0", "0", "1", "12", "60", "130", "420", "8400", "101080", "781200", "4435200", "37714600", "607807200", "8660652000", "94007313400", "914497584000", "11566931376000", "198256136478400", "3275456501116800", "46558791351072000", "636647461257808000", "10238792220969312000", "194852563745775936000" ]	Expansion of e.g.f. exp(x^3/6 * (1+x)^3).
A361569	[ "1", "0", "0", "0", "1", "20", "180", "840", "1715", "2520", "88200", "1940400", "29111775", "303603300", "2188286100", "12549537000", "143029511625", "3397035642000", "71419225878000", "1170096883956000", "15075357741068625", "163540869094102500", "2025016641129982500", "40912918773391665000" ]	Expansion of e.g.f. exp(x^4/24 * (1+x)^4).
A361570	[ "1", "0", "2", "12", "36", "240", "2280", "15120", "122640", "1330560", "13335840", "136382400", "1657212480", "20860519680", "262278656640", "3585207225600", "52249374777600", "772773281280000", "11907924610982400", "193962388523904000", "3253343368231756800", "56051640629816832000" ]	Expansion of e.g.f. exp( (x * (1+x))^2 ).
A361571	[ "1", "0", "0", "6", "72", "360", "1080", "15120", "302400", "3689280", "32659200", "359251200", "6965481600", "133880947200", "2070484416000", "30305353478400", "559684629504000", "12582442768896000", "271843009108070400", "5401042458152140800", "111578968350001152000", "2657164887872022528000" ]	Expansion of e.g.f. exp( (x * (1+x))^3 ).
A361572	[ "1", "0", "0", "6", "72", "720", "7560", "90720", "1270080", "20381760", "364694400", "7125148800", "150186960000", "3393726336000", "81882210009600", "2102315389574400", "57244753133568000", "1647544166940672000", "49957730917981286400", "1591303422125646028800" ]	Expansion of e.g.f. exp( (x / (1-x))^3 ).
A361573	[ "1", "0", "0", "1", "12", "120", "1210", "13020", "152880", "1975960", "28148400", "440470800", "7525441000", "139375236000", "2778421245600", "59239029249400", "1343609515248000", "32274288638592000", "818014942318974400", "21809788600885084800", "610079100418595808000", "17863467401461938256000" ]	Expansion of e.g.f. exp(x^3/(6 * (1 - x)^3)).
A361574	[ "1", "3", "8", "21", "68", "242", "861", "3151", "11874", "45192", "173496", "673042" ]	a(n) is the number of Fibonacci meanders of length m*n and central angle 360/m degrees where m = 3.
A361576	[ "1", "0", "0", "0", "24", "480", "7200", "100800", "1431360", "21772800", "370137600", "7185024000", "158150361600", "3848298854400", "100865282918400", "2799294930432000", "81599752346112000", "2492894621048832000", "79852538982408192000", "2684220785621286912000" ]	Expansion of e.g.f. exp( (x / (1-x))^4 ).
A361577	[ "1", "0", "0", "0", "1", "20", "300", "4200", "58835", "849240", "12814200", "203742000", "3430355775", "61363001700", "1168815948300", "23734579869000", "513878948207625", "11850279026586000", "290440507342986000", "7543064638441332000", "206860683821114948625", "5968372055889205462500" ]	Expansion of e.g.f. exp(x^4/(24 * (1 - x)^4)).
A361578	[ "1", "0", "1", "1", "5", "8", "30", "85", "382", "1550", "7352" ]	Number of 5-connected polyhedra (or 5-connected simple planar graphs) with n nodes
A361579	[ "1", "0", "1", "0", "3", "1", "0", "51", "12", "1", "0", "3614", "447", "34", "1", "0", "991930", "53675", "2885", "85", "1", "0", "1051469032", "21514470", "741455", "16665", "201", "1", "0", "4366988803688", "30405612790", "642187105", "9816380", "90678", "462", "1", "0", "71895397383029040", "160152273169644", "2024633081100", "19625842425", "122330544", "474138", "1044", "1" ]	Triangular array read by rows. T(n,k) is the number of labeled digraphs on [n] with exactly k source-like components, n >= 0, 0 <= k <= n.
A361580	[ "1", "2", "3", "2", "5", "32", "7", "42", "3", "52", "11", "6432", "13", "72", "53", "842", "17", "9632", "19", "10542", "73", "112", "23", "1286432", "5", "132", "93", "14742", "29", "15106532", "31", "16842", "113", "172", "75", "181296432", "37", "192", "133", "20108542", "41", "21147632", "43", "221142", "15953", "232", "47", "24161286432", "7", "251052" ]	If n is composite, replace n with the concatenation of its nontrivial divisors, written in decreasing order, each divisor being written in base 10 with its digits in normal order, otherwise a(n) = n.
A361581	[ "1", "2", "3", "2", "5", "32", "7", "42", "3", "52", "11", "6432", "13", "72", "53", "842", "17", "9632", "19", "1542", "73", "112", "23", "2186432", "5", "312", "93", "41742", "29", "51016532", "31", "61842", "113", "712", "75", "812196432", "37", "912", "313", "2018542", "41", "12417632", "43", "221142", "51953", "322", "47", "42612186432", "7", "520152" ]	If n is composite, replace n with the concatenation of its nontrivial divisors, written in decreasing order, each divisor being written in base 10 with its digits in reverse order, otherwise a(n) = n.
A361582	[ "1", "0", "1", "0", "1", "2", "0", "5", "5", "6", "0", "83", "62", "42", "31", "0", "5048", "2494", "1172", "592", "302", "0", "1047008", "330063", "103961", "38312", "15616", "5984", "0", "705422362", "137934757", "28095923", "7243110", "2297690", "795930", "243668", "0", "1580348371788", "184557780045", "23226116293", "3951426731", "914429926", "261269562", "79512478", "20286025" ]	Triangle read by rows: T(n,k) is the number of digraphs on n unlabeled nodes with k strongly connected components.
A361583	[ "1", "1", "3", "12", "88", "1217", "34672", "2039085", "246005109", "60296886108", "29828186693218", "29663937774464786", "59172529527454608139", "236453014376786629601848", "1891427400988740573006253862", "30274661556583530830890359188257", "969429810937979825934973090455224882" ]	Number of digraphs on n unlabeled nodes whose strongly connected components are complete digraphs.
A361584	[ "1", "1", "3", "12", "88", "1239", "36540", "2226595", "277421616", "69974281748", "35535207035048", "36224521019293188", "74004483908461354689", "302712665772844097945072", "2477999475270966827490305948", "40583406022745170376459610683073", "1329552679157905406495248763876363056" ]	Number of digraphs on n unlabeled nodes whose strongly connected components are directed cycles or single vertices.
A361585	[ "1", "0", "1", "1", "8", "28", "736", "17879", "1568614", "196581247", "62857465075", "34431266945361", "42146672798547398", "95881304594606248756", "459546334152150732106700", "4253461062245855670436620669", "80700118619568448244440535541825", "3011390106783578987361705575335328331" ]	Number of digraphs on n unlabeled nodes whose strongly connected components are directed cycles.
A361586	[ "1", "0", "1", "5", "90", "5289", "1071691", "712342075", "1585944117738", "12152982231404393", "328276896613548366675", "31834464336872565979301363", "11234630426387288679040317490771", "14576388456695908232721134339830232699", "70075904005979773819582865772534172929477101" ]	Number of directed graphs on n unlabeled nodes in which every node belongs to a directed cycle.
A361587	[ "1", "0", "1", "0", "1", "1", "0", "5", "4", "4", "0", "83", "56", "36", "24", "0", "5048", "2406", "1101", "542", "267", "0", "1047008", "324917", "101307", "37017", "14947", "5647", "0", "705422362", "136882286", "27757789", "7134897", "2257234", "779257", "237317", "0", "1580348371788", "183851281949", "23086772643", "3922864504", "907027520", "258909828", "78691767", "20035307" ]	Triangle read by rows: T(n,k) is the number of weakly connected digraphs on n unlabeled nodes with k strongly connected components.
A361588	[ "1", "0", "0", "0", "1", "1", "0", "5", "4", "4", "0", "83", "57", "37", "25", "0", "5048", "2411", "1110", "550", "271", "0", "1047008", "325015", "101467", "37140", "15024", "5682", "0", "705422362", "136887749", "27765860", "7139149", "2259378", "780314", "237684", "0", "1580348371788", "183852357683", "23088181536", "3923330808", "907186816", "258971872", "78716548", "20042357" ]	Triangle read by rows: T(n,k) is the number of digraphs on n unlabeled nodes with k strongly connected components and without isolated nodes.
A361589	[ "1", "0", "1", "4", "25", "271", "5682", "237684", "20042357", "3404651985", "1162523674892", "796395726736678", "1093229314594543016", "3004753338859186373234", "16527845763725396055765240", "181891586856152393087373330332", "4004313490358484085907684748704180", "176328671349936542115174881107633828418" ]	Number of acyclic digraphs on n unlabeled nodes without isolated nodes.
A361590	[ "1", "0", "1", "1", "0", "2", "5", "5", "0", "6", "90", "55", "42", "0", "31", "5289", "2451", "974", "592", "0", "302", "1071691", "323709", "94332", "29612", "15616", "0", "5984", "712342075", "135208025", "25734232", "6059018", "1650492", "795930", "0", "243668", "1585944117738", "181427072519", "21650983294", "3358042412", "704602272", "174576110", "79512478", "0", "20286025" ]	Triangle read by rows: T(n,k) is the number of digraphs on n unlabeled nodes with exactly k strongly connected components of size 1.
A361592	[ "1", "0", "1", "1", "0", "3", "18", "21", "0", "25", "1699", "1080", "774", "0", "543", "587940", "267665", "103860", "59830", "0", "29281", "750744901", "225144360", "64169325", "19791000", "10110735", "0", "3781503", "3556390155318", "672637205149", "126726655860", "29445913175", "7939815030", "3767987307", "0", "1138779265" ]	Triangular array read by rows. T(n,k) is the number of labeled digraphs on [n] with exactly k strongly connected components of size 1, n>=0, 0<=k<=n.

A361468

[ "1", "12", "30", "117", "56", "40", "132", "1080", "775", "672", "182", "390", "306", "176", "1680", "9801", "380", "9300", "552", "6552", "3960", "2184", "870", "144", "2793", "408", "19500", "1716", "992", "2240", "1406", "88452", "5460", "4560", "7392", "90675", "1722", "736", "9180", "60480", "1892", "5280", "2256", "126", "43400", "1160", "2862", "32670", "16093", "3724", "456", "442", "3540", "26000" ]

a(n) = A249670(A003961(n)).

A361469

[ "0", "3", "3", "3", "4", "4", "4", "7", "3", "7", "3", "4", "4", "5", "7", "6", "4", "6", "5", "7", "7", "6", "4", "6", "4", "5", "7", "5", "6", "8", "3", "9", "6", "7", "8", "6", "4", "6", "7", "11", "4", "8", "6", "4", "7", "5", "5", "7", "4", "5", "5", "3", "5", "8", "5", "9", "8", "9", "3", "8", "4", "6", "7", "7", "8", "7", "6", "7", "5", "9", "3", "8", "6", "5", "7", "6", "7", "8", "5", "10", "6", "7", "5", "6", "8", "7", "9", "10", "4", "10", "8", "5", "6", "6", "9", "10", "4", "7", "6", "5", "5", "6", "6", "7", "11" ]

a(n) = bigomega(A249670(A003961(n))).

A361473

[ "1", "10", "27", "45", "143", "306", "903", "465", "1215", "3037", "2418", "4809", "17193", "8349", "32055", "75847", "117705", "306075", "379395", "467955", "1269075", "2517687", "1809295", "4720023", "6375915", "12961575", "21540987", "35647010", "16615305", "192717405", "268822806", "186269391", "247067415" ]

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

A361475

[ "0", "1", "0", "3", "1", "0", "7", "4", "1", "0", "15", "13", "5", "1", "0", "31", "40", "21", "6", "1", "0", "63", "121", "85", "31", "7", "1", "0", "127", "364", "341", "156", "43", "8", "1", "0", "255", "1093", "1365", "781", "259", "57", "9", "1", "0", "511", "3280", "5461", "3906", "1555", "400", "73", "10", "1", "0", "1023", "9841", "21845", "19531", "9331", "2801", "585", "91", "11", "1", "0" ]

Array read by ascending antidiagonals: A(n, k) = (k^n - 1)/(k - 1), with k >= 2.

A361476

[ "0", "1", "4", "12", "34", "99", "308", "1040", "3820", "15197", "65060", "297828", "1449742", "7468527", "40555732", "231335944", "1381989864", "8623700793", "56078446596", "379233142780", "2662013133274", "19362917621979", "145719550012276", "1133023004941248", "9090156910550084", "75161929739797493", "639793220877941476" ]

Antidiagonal sums of A361475.

A361477

[ "1", "1", "1", "1", "2", "1", "2", "1", "2", "3", "1", "3", "1", "3", "2", "1", "2", "3", "4", "3", "4", "1", "4", "3", "2", "3", "4", "3", "2", "3", "2", "1", "2", "3", "4", "6", "6", "5", "6", "6", "4", "5", "1", "5", "6", "5", "4", "3", "2", "6", "6", "1", "6", "5", "6", "6", "1", "6", "4", "6", "2", "3", "2", "1", "2", "3", "4", "6", "12", "5", "12", "3", "12", "10", "6", "10", "4", "10", "12", "6", "4", "5", "6", "10" ]

a(n) is the number of integers whose binary expansions have the same multiset of run-lengths as that of n.

A361478

[ "0", "1", "2", "3", "4", "6", "5", "4", "6", "7", "8", "14", "9", "11", "13", "10", "9", "11", "13", "12", "9", "11", "13", "8", "14", "15", "16", "30", "17", "23", "29", "18", "20", "22", "26", "19", "25", "27", "18", "20", "22", "26", "21", "18", "20", "22", "26", "17", "23", "29", "24", "28", "19", "25", "27", "18", "20", "22", "26", "19", "25", "27", "24", "28", "17", "23", "29", "16", "30" ]

Irregular table T(n, k), n >= 0, k = 1..A361477(n), read by rows; the n-th row lists the integers whose binary expansions have the same multiset of run-lengths as that of n.

A361479

[ "0", "1", "2", "3", "4", "5", "4", "7", "8", "9", "10", "9", "12", "9", "8", "15", "16", "17", "18", "19", "18", "21", "18", "17", "24", "19", "18", "19", "24", "17", "16", "31", "32", "33", "34", "35", "36", "37", "36", "35", "34", "37", "42", "37", "36", "37", "34", "33", "48", "35", "36", "51", "36", "37", "36", "35", "56", "35", "34", "35", "48", "33", "32", "63", "64", "65", "66", "67" ]

a(n) is the least integer whose binary expansion has the same multiset of run-lengths as that of n.

A361480

[ "0", "1", "2", "3", "6", "5", "6", "7", "14", "13", "10", "13", "12", "13", "14", "15", "30", "29", "26", "27", "26", "21", "26", "29", "28", "27", "26", "27", "28", "29", "30", "31", "62", "61", "58", "59", "54", "53", "54", "59", "58", "53", "42", "53", "54", "53", "58", "61", "60", "59", "54", "51", "54", "53", "54", "59", "56", "59", "58", "59", "60", "61", "62", "63", "126", "125" ]

a(n) is the greatest integer whose binary expansion has the same multiset of run-lengths as that of n.

A361481

[ "0", "1", "2", "3", "4", "6", "5", "7", "8", "14", "9", "11", "13", "10", "12", "15", "16", "30", "17", "23", "29", "18", "20", "22", "26", "19", "25", "27", "21", "24", "28", "31", "32", "62", "33", "47", "61", "34", "40", "46", "58", "35", "39", "49", "55", "57", "59", "36", "38", "44", "50", "52", "54", "37", "41", "43", "45", "53", "42", "48", "60", "51", "56", "63", "64", "126", "65" ]

Distinct values of A361478, in order of appearance.

A361482

[ "0", "1", "2", "3", "4", "6", "5", "7", "8", "10", "13", "11", "14", "12", "9", "15", "16", "18", "21", "25", "22", "28", "23", "19", "29", "26", "24", "27", "30", "20", "17", "31", "32", "34", "37", "41", "47", "53", "48", "42", "38", "54", "58", "55", "49", "56", "39", "35", "59", "43", "50", "61", "51", "57", "52", "44", "62", "45", "40", "46", "60", "36", "33", "63", "64", "66", "69", "73" ]

Inverse permutation to A361481.

A361483

[ "7", "13", "37", "61", "97", "103", "127", "163", "193", "211", "223", "307", "313", "331", "337", "397", "421", "463", "487", "541", "571", "601", "607", "631", "673", "691", "727", "757", "853", "907", "937", "967", "1021", "1033", "1051", "1063", "1117", "1153", "1171", "1231", "1237", "1297", "1303", "1327", "1381", "1453", "1531", "1567", "1621", "1657", "1693", "1723" ]

Primes p such that p + 256 is also prime.

A361484

[ "11", "29", "59", "89", "101", "107", "131", "149", "179", "197", "227", "239", "257", "311", "317", "347", "479", "509", "521", "557", "617", "641", "659", "701", "719", "809", "887", "911", "941", "947", "971", "977", "1019", "1031", "1097", "1109", "1151", "1181", "1187", "1229", "1277", "1289", "1319", "1361", "1367", "1439", "1481", "1487", "1499", "1571", "1601" ]

Primes p such that p + 512 is also prime.

A361485

[ "7", "37", "67", "73", "79", "127", "139", "157", "163", "193", "199", "277", "283", "337", "349", "409", "457", "463", "487", "499", "547", "577", "613", "643", "673", "709", "787", "823", "853", "877", "883", "907", "1039", "1063", "1087", "1117", "1129", "1213", "1249", "1327", "1399", "1423", "1453", "1567", "1597", "1609", "1663", "1669", "1753", "1777", "1873", "1879" ]

Primes p such that p + 1024 is also prime.

A361486

[ "1", "1", "1", "1", "2", "2", "3", "2", "2", "3", "2", "2", "3", "2", "2", "3", "1", "3", "3", "1", "4", "1", "4", "3", "5", "5", "1", "4", "3", "4", "5", "4", "4", "5", "6", "6", "7", "4", "4", "5", "5", "6", "2", "4", "1", "4", "5", "1", "6", "2", "6", "4", "6", "5", "5", "7", "2", "3", "4", "6", "5", "5", "7", "2", "3", "8", "1", "4", "3", "6", "7", "5", "5", "3", "5", "7", "6", "3", "1", "1", "7", "8", "7", "7", "4", "5", "8", "5", "9", "6", "6", "8", "7", "7", "6", "8", "9", "9", "3" ]

Lexicographically earliest sequence of positive numbers on a square spiral such that no three equal numbers are collinear.

A361488

[ "1", "0", "0", "2", "2", "0", "6", "12", "6", "20", "60", "60", "90", "280", "420", "532", "1330", "2520", "3444", "6804", "14112", "21912", "37884", "77616", "133914", "223080", "432432", "793364", "1341912", "2471040", "4629196", "8076640", "14453010", "26960232", "48308832", "85794852", "157947816", "287413152", "512697900", "933072064" ]

Diagonal of rational function 1/(1 - (x^3 + y^3 + x^4*y)).

A361489

[ "1", "1", "2", "6", "19", "72", "313", "1472", "7612", "42679", "255515", "1632710", "11065057", "79065807", "594174922", "4679473130", "38500353667", "330172915164", "2944613004359", "27253908250340", "261328607398332", "2591724561444621", "26545170005412613", "280411070646125638" ]

Expansion of e.g.f. exp(exp(x) - 1 + x^3/6).

A361490

[ "8", "45", "52", "75", "92", "99", "130", "147", "164", "195", "236", "255", "266", "333", "406", "423", "430", "477", "494", "555", "574", "627", "670", "711", "716", "777", "782", "801", "806", "903", "908", "915", "932", "935", "938", "969", "1010", "1017", "1022", "1065", "1076", "1233", "1244", "1443", "1474", "1479", "1490", "1533", "1556", "1635", "1724", "1737", "1790", "1833", "1844", "2007", "2012" ]

a(1) = 8; for n > 1, a(n) is the least triprime > a(n-1) such that a(n) - a(n-1) and a(n) + a(n-1) are both prime.

A361491

[ "1", "73", "2521", "85681", "2910673", "98877241", "3358915561", "114104251873", "3876185648161", "131676207785641", "4473114879063673", "151954229680379281", "5161970694253831921", "175355049374949906073", "5956909708054042974601", "202359575024462511230401", "6874268641123671338859073", "233522774223180363009978121" ]

Expansion of x*(1+38*x+x^2)/((1-x)*(x^2-34*x+1)).

A361492

[ "1", "1", "2", "6", "30", "30", "210", "210", "210", "17430", "30030", "60060", "510510", "3573570" ]

Common difference corresponding to increasing arithmetic progression of at least n >= 2 primes whose first term is A284708(n); a(1) = 1.

A361493

[ "1", "1", "2", "11", "39", "172", "1163", "6547", "41772", "335139", "2486215", "20078610", "186139957", "1676540257", "16077206122", "168739976555", "1763716943267", "19358116589964", "226362412711759", "2669223655597955", "32748447519013132", "421204995451111971", "5496921281576148363" ]

Expansion of e.g.f. exp(exp(x) - 1 + x^3).

A361497

[ "3", "4", "4", "4", "7", "7", "6", "8", "6", "10", "12", "6", "11", "8", "12", "14", "10", "17", "14", "16", "16", "16", "15", "8", "20", "21", "10", "18", "14", "24", "18", "20", "12", "16", "22", "25", "30", "20", "22", "12", "32", "36", "12", "30", "16", "42", "22", "32", "16", "22", "30", "31", "32", "22", "26", "12", "36", "39", "18", "37", "28", "48", "35", "28", "32", "42", "32", "34", "46", "30", "29", "26", "40", "49", "52", "14", "45", "24", "34", "48", "32", "56", "30", "44", "38", "40", "36", "45", "20", "54", "74", "22", "49", "28" ]

Number of cusps in Prym-Teichmuller curve W_D(4) of discriminant D = A361169(n).

A361498

[ "0", "0", "0", "0", "0", "0", "0", "0", "0", "1", "1", "1", "1", "2", "1", "1", "2", "1", "1", "3", "5", "1", "4", "4", "6", "4", "5", "7", "5", "9", "6", "9", "8", "9", "10", "9", "8", "9", "9", "9", "11", "14", "10", "11", "14", "15", "16", "14", "15", "14", "21", "18", "22", "19", "19", "17", "23", "21", "17", "23", "19", "27", "22", "30", "18", "27", "22", "34", "30", "25", "29", "22", "41", "35", "43", "26", "36", "29", "33", "36", "30", "39", "35", "39", "43", "55", "37", "40", "38", "49", "46", "38", "44", "40" ]

Genus of Prym-Teichmuller curve W_D(4) of discriminant D = A361169(n).

A361499

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

Number of orbifold points of order 2 in Prym-Teichmuller curve W_D(4) of discriminant D = A361169(n).

A361500

[ "1", "0", "0", "2", "0", "0", "2", "1", "2", "0", "0", "2", "1", "0", "2", "0", "0", "2", "2", "0", "0", "4", "1", "4", "0", "2", "2", "0", "0", "0", "3", "0", "2", "2", "0", "1", "0", "2", "3", "4", "2", "0", "4", "0", "2", "0", "2", "0", "0", "6", "0", "0", "0", "4", "2", "4", "0", "4", "4", "1", "0", "0", "2", "0", "0", "4", "4", "0", "0", "6", "5", "2", "0", "2", "0", "6", "0", "0", "3", "0", "2", "2", "6", "0", "2", "0", "4", "3", "4", "0", "0", "6", "0", "2" ]

Number of orbifold points of order 3 in Prym-Teichmuller curve W_D(4) of discriminant D = A361169(n).

A361501

[ "11", "112", "1124", "11248", "1124816", "112481623", "11248162328", "1124816232838", "112481623283849", "11248162328384962", "1124816232838496270", "112481623283849627077", "2486232838496270779", "248623283849627077997", "248623283849627077997113", "248623283849627077997113118", "248623283849627077997113118128" ]

A variant of A359143 in which all copies of a digit d are erased only when d is both the leading digit and the final digit of (a(n) concatenated with sum of digits of a(n)).

A361502

[ "2", "3", "4", "8", "13", "42", "347", "3466", "49012", "528231", "717126", "63056215", "1375559400", "7038527851" ]

Index of n-th prime in A359804.

A361503

[ "2", "3", "5", "2", "3", "5", "7", "3", "2", "5", "7", "11", "3", "5", "7", "11", "5", "3", "7", "5", "11", "7", "2", "3", "5", "7", "11", "5", "7", "2", "3", "5", "7", "3", "5", "7", "2", "5", "7", "11", "13", "5", "2", "3", "5", "2", "7", "3", "5", "7", "11", "5", "3", "2", "5", "7", "11", "2", "5", "3", "7", "5", "3", "7", "2", "11", "3", "7", "5", "3", "7", "5", "11", "7", "5", "11", "7", "5", "3", "7", "11", "3", "2", "5", "7", "2", "5", "3", "2", "5", "7", "13", "3", "5", "2", "3" ]

a(1)=2; thereafter a(n) = smallest prime that does not divide b(n-1)*b(n), where b(k) = A359804(k).

A361504

[ "1", "2", "3", "5", "4", "6", "8", "10", "9", "7", "13", "14", "42", "12", "11", "24", "347", "19", "3466", "15", "16", "17", "49012", "25", "18", "31", "32", "20", "528231", "21", "717126", "38", "22", "44", "23", "35", "63056215", "47", "45", "26", "1375559400", "27", "7038527851", "28", "29", "55" ]

Index of n in A359804, or -1 if n never appears there.

A361505

[ "1", "2", "5", "10", "24", "38", "87", "172", "349", "706", "1407", "2752", "5487", "11103", "22285", "44429", "88993", "177746", "356460", "712129", "1425163", "2849424", "5701776", "11401709", "22804522", "45608572", "91219022", "182438457", "364879209", "729757797", "1459514883", "2919031155", "5838065175", "11676129412", "23352260426" ]

Index of 2^n in A359804.

A361506

[ "1", "3", "8", "19", "45", "102", "232", "524", "1181", "2659", "5984", "13466", "30300", "68177", "153400", "345151", "776590", "1747330", "3931495", "8845865", "19903197", "44782195", "100759940", "226709865", "510097199", "1147718700", "2582367075", "5810325920", "13073233320", "29414774970", "66183243690", "148912298300", "335052671200", "753868510200" ]

a(n) = floor( (4/5)*( (9/4)^(n+1)-1 ) ).

A361507

[ "1", "3", "7", "16", "37", "84", "190", "428", "964", "2170", "4883", "10987", "24721", "55623", "125152", "281593", "633585", "1425567", "3207526", "7216934", "16238102", "36535730", "82205393", "184962135", "416164804", "936370810", "2106834323", "4740377227", "10665848761", "23998159713", "53995859355", "121490683549", "273354037986", "615046585469", "1383854817306", "3113673338939" ]

a(0) = 1; thereafter a(n) = floor((9/4)*a(n-1)) + 1.

A361517

[ "3", "4", "5", "11", "17", "27", "35", "37", "49", "59", "69", "81", "91", "103", "115", "123", "135", "137", "167", "175", "189", "199", "207", "287", "295", "307", "361", "1051", "2507", "2757", "2917", "3057", "3081", "7255", "7361", "7871", "16173" ]

The value of n for which the two-player impartial {0,1}-Toggle game on a generalized Petersen graph GP(n,2) with a (1,0)-weight assignment is a next-player winning game.

A361521

[ "0", "0", "0", "0", "2", "0", "0", "5", "4", "0", "0", "9", "12", "6", "0", "0", "14", "24", "21", "8", "0", "0", "20", "40", "45", "32", "10", "0", "0", "27", "60", "78", "72", "45", "12", "0", "0", "35", "84", "120", "128", "105", "60", "14", "0", "0", "44", "112", "171", "200", "190", "144", "77", "16", "0", "0", "54", "144", "231", "288", "300", "264", "189", "96", "18", "0" ]

Array read by descending antidiagonals. A(n, k) is the number of the nonempty multiset combinations of {0, 1} as defined in A361682.

A361522

[ "1", "0", "1", "0", "2", "0", "6", "0", "24", "0", "120", "0", "720", "0", "5040", "0", "40320", "0", "362880", "0", "3628800", "0", "39916800", "0", "479001600", "0", "6227020800", "0", "87178291200", "0", "1307674368000", "0", "20922789888000", "0", "355687428096000", "0", "6402373705728000", "0", "121645100408832000", "0", "2432902008176640000" ]

The aerated factorial numbers.

A361523

[ "1", "1", "1", "1", "2", "4", "1", "3", "17", "54", "1", "6", "61", "892", "9235", "1", "10", "220", "8159", "406653", "10538496" ]

Triangle read by rows: T(n,k) is the number of ways of dividing an n X k rectangle into integer-sided rectangles, up to rotations and reflections.

A361524

[ "1", "1", "4", "54", "9235", "10538496" ]

Number of ways of dividing an n X n square into integer-sided rectangles, up to rotations and reflections.

A361525

[ "1", "3", "17", "54", "892", "8159", "80021", "791821", "7906439", "79069308" ]

Number of ways of dividing an n X 3 rectangle into integer-sided rectangles, up to rotations and reflections.

A361526

[ "1", "6", "61", "892", "9235", "406653", "9252097", "211703640" ]

Number of ways of dividing an n X 4 rectangle into integer-sided rectangles, up to rotations and reflections.

A361527

[ "1", "0", "1", "0", "1", "3", "0", "2", "21", "25", "0", "6", "213", "774", "543", "0", "24", "3470", "30275", "59830", "29281", "0", "120", "95982", "1847265", "7757355", "10110735", "3781503", "0", "720", "4578588", "190855000", "1522899105", "3944546095", "3767987307", "1138779265" ]

Triangular array read by rows. T(n,k) is the number of labeled digraphs on [n] having exactly k strongly connected components all of which are simple cycles, n >= 0, 0 <= k <= n.

A361528

[ "1", "1", "8", "75", "804", "9681", "129168", "1889379", "30037500", "515342817", "9484627608", "186305208219", "3888697965012", "85920579594225", "2002828537732896", "49107722192594739", "1263165207424720812", "34004577057249890241", "955970215914084949800", "28011115058953357075563", "853924857091970071203972" ]

a(n) = (2+n)*(2*a(n-1) - (n-2)*a(n-2)) with a(0)=a(1)=1.

A361530

[ "23", "37", "53", "73", "113", "127", "131", "137", "139", "151", "157", "173", "179", "193", "197", "211", "223", "229", "233", "239", "241", "271", "283", "293", "311", "313", "317", "331", "337", "347", "353", "359", "367", "373", "379", "383", "389", "397", "421", "431", "433", "457", "523", "541", "547", "571", "593", "613", "617", "631", "673", "677", "719" ]

Primes that can be written as the result of shuffling the decimal digits of two primes.

A361531

[ "1", "-1", "0", "2", "-3", "-2", "21", "-44", "-62", "631", "-1367", "-3170", "34849", "-86855", "-302964", "3058342", "-8509971", "-36488802", "430842051", "-1111575888", "-6244999438", "78663444549", "-250850311489", "-1724880111306", "18475299723737", "-65061274823853", "-444914618968648", "6831921081061986" ]

Expansion of e.g.f. exp(1 - exp(x) + x^3/6).

A361532

[ "1", "1", "4", "19", "118", "886", "7786", "78184", "881644", "11017108", "150966856", "2249261356", "36181351504", "624658612384", "11516406883528", "225740649754936", "4686671645814736", "102712289940757264", "2369128149877075264", "57359541280704038128", "1454229915957292684576" ]

Expansion of e.g.f. exp((x + x^2/2)/(1-x)).

A361533

[ "1", "0", "0", "1", "4", "20", "130", "980", "8400", "80920", "865200", "10164000", "130114600", "1802600800", "26867640800", "428661633400", "7288513232000", "131558835408000", "2512282795422400", "50600743739145600", "1071998968264224000", "23829055696093648000", "554524256514356128000" ]

Expansion of e.g.f. exp(x^3/(6 * (1-x))).

A361535

[ "1", "10", "61", "290", "1172", "4212", "13833", "42262", "121625", "332764", "871641", "2197936", "5359005", "12679730", "29200593", "65617892", "144189054", "310400110", "655669910", "1360910666", "2779007594", "5589070978", "11081585154", "21679798590", "41883282555", "79958881544", "150943109191", "281926365224" ]

Expansion of g.f. 1 / Product_{n>=1} ((1 - x^n)^6 * (1 - x^(2*n-1))^4).

A361536

[ "1", "3", "60", "2037", "92187", "5066952", "322801089", "23197971285", "1848188250810", "161297106209607", "15285968218925460", "1562519987561305566", "171348519312001997550", "20068058089211306151393", "2500498134501774994768119", "330350627790472265384885061", "46136067767500181432129130897" ]

Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A(x)^(3*n) / n!.

A361537

[ "1", "3", "75", "3234", "186471", "13063908", "1060481214", "97053553710", "9840717984447", "1092337371705273", "131589391554509112", "17089208887923714204", "2379797411747290723350", "353790840030976298935989", "55935780589531899802966062", "9373903063348266793396858620" ]

Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A(x)^(4*n) / n!.

A361538

[ "1", "5", "244", "19090", "1839075", "199363606", "23320604384", "2876887986028", "369107300988219", "48807370374419910", "6610055144592717246", "912769451975745598299", "128087096344387852459658", "18219369599509083643661044", "2621701165622760015979876800", "381039123615762485580377457688" ]

Central terms of triangle A361050.

A361539

[ "3", "39", "426", "4550", "50085", "577731", "7022596", "90148860", "1222753815", "17515226465", "264663151038", "4212100028994", "70475063838361", "1237144088015535", "22735980569119560", "436467520716475064", "8733235757816095083", "181740089309026259565", "3925458146197916823970" ]

a(n) = A361540(n, n-2) for n >= 2, a diagonal of triangle A361540.

A361540

[ "1", "1", "1", "3", "4", "1", "22", "39", "18", "1", "269", "604", "426", "92", "1", "4616", "12625", "12040", "4550", "520", "1", "102847", "332766", "401355", "218300", "50085", "3222", "1", "2824816", "10574725", "15456756", "11017895", "3867080", "577731", "21700", "1", "92355769", "393171416", "676130644", "597596216", "284455150", "69038984", "7022596", "157544", "1" ]

Expansion of e.g.f. A(x,y) satisfying A(x,y) = Sum_{n>=0} (A(x,y)^n + y)^n * x^n/n!, as a triangle read by rows.

A361541

[ "1", "4", "56", "1220", "34788", "1203152", "48418384", "2210163032", "112501779300", "6308565897088", "386149471644704", "25614932030415636", "1830512170952711968", "140224558208217547440", "11464991752291729651224", "996723500374559386157920", "91824970792933898453830680" ]

Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(4*n) * A(x)^n / n!.

A361542

[ "1", "4", "84", "2940", "137228", "7809680", "517517212", "38860889496", "3248881861500", "298704250964336", "29928006672383280", "3244628959712243628", "378449007991303855532", "47261928190105905687600", "6293239981401396941576632", "890249832854933140207681360", "133355904852469516343820132852" ]

Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(4*n) * A(x)^(2*n) / n!.

A361543

[ "1", "4", "112", "5380", "346788", "27285968", "2498963752", "259124694312", "29885849525700", "3786931724896768", "522451837498888672", "77929657518224116484", "12496899169394954817144", "2144326582901160246138160", "392104633203721656029928184", "76134826269461672101153285664" ]

Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(4*n) * A(x)^(3*n) / n!.

A361544

[ "1", "4", "39", "604", "12625", "332766", "10574725", "393171416", "16744363569", "803841993370", "42957812253301", "2529951235854516", "162852898603253209", "11378885054925777494", "858009440175419213445", "69471138931959493061296", "6013997809048628612191585", "554545575488282609142617778" ]

a(n) = A361540(n,1) for n >= 1, a column of triangle A361540.

A361545

[ "1", "0", "0", "0", "1", "5", "30", "210", "1715", "15750", "160650", "1801800", "22043175", "292116825", "4168464300", "63725161500", "1039028615625", "17998106626500", "330068683444500", "6388785205803000", "130156170633113625", "2783924007745505625", "62375052003905891250", "1460924768552182683750" ]

Expansion of e.g.f. exp(x^4/(24 * (1-x))).

A361546

[ "1", "5", "3", "5", "9", "5", "9", "11", "45", "23", "35", "15", "3", "9", "27", "51", "27", "53", "9", "39", "23", "249", "51", "51", "131", "221", "29", "105", "321", "179", "5", "221", "111", "411", "191", "65", "83", "75", "95", "101", "147", "83", "149", "111", "203", "131", "9", "245", "281", "15", "83", "65", "299", "39", "51", "51", "225", "65", "81", "125", "611", "143", "65", "107", "21" ]

a(n) is the least odd number k such that k*2^prime(n) + 1 is prime, or -1 if no such number k exists.

A361547

[ "1", "0", "0", "0", "0", "1", "6", "42", "336", "3024", "30366", "335412", "4041576", "52756704", "741620880", "11169844686", "179448036768", "3063069801792", "55360031126400", "1056123043335360", "21208345049147256", "447183762148547424", "9877939209960101280", "228112734232663600320" ]

Expansion of e.g.f. exp(x^5/(120 * (1-x))).

A361548

[ "1", "1", "4", "20", "126", "966", "8656", "88544", "1016380", "12920156", "179996816", "2725070096", "44521522024", "780344770440", "14599772973696", "290311643773376", "6112190642062096", "135798496839920144", "3174483084427144000", "77872118431269176896", "1999809157085214044896" ]

Expansion of e.g.f. exp((x + x^2/2 + x^3/6)/(1-x)).

A361549

[ "1", "18", "426", "12040", "401355", "15456756", "676130644", "33151425840", "1802216703285", "107652497473180", "7012494336544686", "494963689847333928", "37648456802884402111", "3071415347513049808740", "267644521958509484952360", "24822151072519637091258976", "2442314922307988498911793385" ]

a(n) = A361540(n,2) for n >= 2, a column of triangle A361540.

A361550

[ "1", "0", "1", "0", "5", "1", "0", "18", "10", "1", "0", "55", "61", "20", "1", "0", "149", "290", "215", "35", "1", "0", "371", "1172", "1660", "555", "56", "1", "0", "867", "4212", "10311", "5850", "1254", "84", "1", "0", "1923", "13833", "54688", "47460", "17773", "2555", "120", "1", "0", "4086", "42262", "256815", "319409", "188300", "46844", "4810", "165", "1", "0", "8374", "121625", "1093790", "1864445", "1621116", "621915", "111348", "8505", "220", "1", "0", "16634", "332764", "4297370", "9717550", "11913160", "6557572", "1818022", "243795", "14290", "286", "1" ]

Expansion of g.f. A(x,y) satisfying x*y = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x,y)^(3*n) - 1/A(x,y)^(3*n+1)), as a triangle read by rows.

A361551

[ "1", "5", "90", "2535", "93840", "4226355", "222038775", "13259599965", "884588496165", "65114097133590", "5239173990133060", "457392343670390700", "43064135370809341250", "4350264113638902544555", "469422682906897831519170", "53897717818214315584719430", "6561919113715122121302125775" ]

Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n (x^(5*n) * A(x)^n) / n!.

A361552

[ "1", "2", "14", "84", "530", "3770", "29446", "240302", "2003914", "17024332", "147306448", "1294859540", "11524690228", "103605031978", "939357512086", "8580744729478", "78898896072996", "729661925134886", "6782435427053490", "63332055630823770", "593793935288453260", "5587934788557993846" ]

Expansion of g.f. A(x) satisfying 2*x = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x)^(3*n) - 1/A(x)^(3*n+1)).

A361553

[ "1", "3", "24", "171", "1335", "11940", "115773", "1160901", "11901537", "124726644", "1332688035", "14455451526", "158660036535", "1758835084221", "19667067522966", "221573079684087", "2512635069594897", "28656903391830291", "328500210705228867", "3782806859877522522", "43738575934977450465" ]

Expansion of g.f. A(x) satisfying 3*x = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x)^(3*n) - 1/A(x)^(3*n+1)).

A361554

[ "1", "4", "36", "296", "2732", "28980", "329996", "3872908", "46575260", "573472248", "7197096168", "91640952360", "1180636398320", "15364364313588", "201691201775092", "2667523242203932", "35510152549696208", "475424653523498396", "6397601663340197268", "86481499341290372804", "1173813146742741571560" ]

Expansion of g.f. A(x) satisfying 4*x = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x)^(3*n) - 1/A(x)^(3*n+1)).

A361555

[ "1", "5", "50", "465", "4925", "59870", "776155", "10364135", "142082065", "1995371980", "28549274995", "414327073520", "6084353526535", "90258375062245", "1350607531232830", "20361436162127965", "308964002231172075", "4715119823819824535", "72324133311820587435", "1114404268419043050750" ]

Expansion of g.f. A(x) satisfying 5*x = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x)^(3*n) - 1/A(x)^(3*n+1)).

A361556

[ "1", "5", "61", "1660", "47460", "1621116", "58002140", "2213389940", "87301563690", "3555890156445", "148125509781095", "6292884402884976", "271565202254735207", "11878392121526009800", "525519782174930309205", "23481280252471520720288", "1058270749214634093475910", "48058678036035725619136698" ]

Central terms of triangle A361550.

A361557

[ "1", "1", "4", "20", "127", "977", "8789", "90267", "1040260", "13275258", "185653535", "2821321725", "46265262553", "813871304989", "15281792484768", "304949014412540", "6442741397501699", "143633948442619765", "3369004776395733829", "82919378806522132407", "2136425765494805888952" ]

Expansion of e.g.f. exp((exp(x) - 1)/(1-x)).

A361558

[ "1", "1", "4", "20", "127", "976", "8776", "90084", "1037555", "13233077", "184956386", "2809098986", "46038214729", "809411443790", "15189361799522", "302932433571356", "6396529241755881", "142523960797017589", "3341115707515530400", "82187749261419720712", "2116421112495023612311" ]

Expansion of e.g.f. exp((x + x^2/2 + x^3/6 + x^4/24)/(1-x)).

A361559

[ "0", "10", "258", "1740", "20070", "48510", "196920", "350370", "937860", "3075030", "4322160", "10641330", "17925180", "22825110", "35827560", "65816010", "113180910", "133937670", "215070570", "288148140", "331474860", "493573080", "633015810", "899599140", "1387338960", "1700082450", "1876303260", "2272556790", "2494333710" ]

a(n) = Sum_{k=1..prime(n)-1} floor(k^5/prime(n)).

A361560

[ "1", "1", "4", "47", "1471", "115042", "21591817", "9455689609", "9464951556046", "21316993121024757", "106689322228222150243", "1174731578884501228621956", "28221161668500867009724237123", "1468937207982284446757761131062629", "164682046577167683717133576752582349216", "39562388056404531283767850863430043742371123" ]

Number of labeled digraphs on [n] all of whose strongly connected components are complete digraphs.

A361562

[ "3", "7", "11", "19", "23", "31", "43", "79", "127", "167", "191", "199", "347", "3539", "5807", "10691", "11279", "12391", "14479", "83339", "117239", "127031", "141079", "269987", "986191", "4031399" ]

Wagstaff numbers that are of the form 4*k + 3.

A361563

[ "5", "13", "17", "61", "101", "313", "701", "1709", "2617", "10501", "42737", "95369", "138937", "267017", "374321" ]

Wagstaff numbers that are of the form 4*k + 1.

A361564

[ "4", "6", "10", "17", "25", "39", "59", "87", "127", "186" ]

Number of (n-3)-connected unlabeled n-node graphs.

A361565

[ "1", "3", "2", "2", "3", "5", "4", "3", "3", "7", "6", "7", "7", "9", "4", "4", "9", "9", "10", "9", "5", "13", "12", "5", "5", "15", "6", "11", "15", "11", "16", "6", "7", "19", "6", "6", "19", "21", "8", "13", "21", "13", "22", "15", "7", "25", "24", "7", "7", "15", "10", "17", "27", "15", "8", "15", "11", "31", "30", "8", "31", "33", "8", "8", "9", "17", "34", "21", "13", "17", "36", "17", "37", "39", "10" ]

a(n) is the numerator of the median of divisors of n.

A361566

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

a(n) is the denominator of the median of divisors of n.

A361567

[ "1", "0", "1", "6", "15", "60", "555", "3150", "17745", "158760", "1399545", "10914750", "102920895", "1104323220", "11249313075", "119330961750", "1426411411425", "17429852840400", "213417453474225", "2791671804271350", "38524272522310575", "537569719902715500", "7732658753799054075" ]

Expansion of e.g.f. exp(x^2/2 * (1+x)^2).

A361568

[ "1", "0", "0", "1", "12", "60", "130", "420", "8400", "101080", "781200", "4435200", "37714600", "607807200", "8660652000", "94007313400", "914497584000", "11566931376000", "198256136478400", "3275456501116800", "46558791351072000", "636647461257808000", "10238792220969312000", "194852563745775936000" ]

Expansion of e.g.f. exp(x^3/6 * (1+x)^3).

A361569

[ "1", "0", "0", "0", "1", "20", "180", "840", "1715", "2520", "88200", "1940400", "29111775", "303603300", "2188286100", "12549537000", "143029511625", "3397035642000", "71419225878000", "1170096883956000", "15075357741068625", "163540869094102500", "2025016641129982500", "40912918773391665000" ]

Expansion of e.g.f. exp(x^4/24 * (1+x)^4).

A361570

[ "1", "0", "2", "12", "36", "240", "2280", "15120", "122640", "1330560", "13335840", "136382400", "1657212480", "20860519680", "262278656640", "3585207225600", "52249374777600", "772773281280000", "11907924610982400", "193962388523904000", "3253343368231756800", "56051640629816832000" ]

Expansion of e.g.f. exp( (x * (1+x))^2 ).

A361571

[ "1", "0", "0", "6", "72", "360", "1080", "15120", "302400", "3689280", "32659200", "359251200", "6965481600", "133880947200", "2070484416000", "30305353478400", "559684629504000", "12582442768896000", "271843009108070400", "5401042458152140800", "111578968350001152000", "2657164887872022528000" ]

Expansion of e.g.f. exp( (x * (1+x))^3 ).

A361572

[ "1", "0", "0", "6", "72", "720", "7560", "90720", "1270080", "20381760", "364694400", "7125148800", "150186960000", "3393726336000", "81882210009600", "2102315389574400", "57244753133568000", "1647544166940672000", "49957730917981286400", "1591303422125646028800" ]

Expansion of e.g.f. exp( (x / (1-x))^3 ).

A361573

[ "1", "0", "0", "1", "12", "120", "1210", "13020", "152880", "1975960", "28148400", "440470800", "7525441000", "139375236000", "2778421245600", "59239029249400", "1343609515248000", "32274288638592000", "818014942318974400", "21809788600885084800", "610079100418595808000", "17863467401461938256000" ]

Expansion of e.g.f. exp(x^3/(6 * (1 - x)^3)).

A361574

[ "1", "3", "8", "21", "68", "242", "861", "3151", "11874", "45192", "173496", "673042" ]

a(n) is the number of Fibonacci meanders of length m*n and central angle 360/m degrees where m = 3.

A361576

[ "1", "0", "0", "0", "24", "480", "7200", "100800", "1431360", "21772800", "370137600", "7185024000", "158150361600", "3848298854400", "100865282918400", "2799294930432000", "81599752346112000", "2492894621048832000", "79852538982408192000", "2684220785621286912000" ]

Expansion of e.g.f. exp( (x / (1-x))^4 ).

A361577

[ "1", "0", "0", "0", "1", "20", "300", "4200", "58835", "849240", "12814200", "203742000", "3430355775", "61363001700", "1168815948300", "23734579869000", "513878948207625", "11850279026586000", "290440507342986000", "7543064638441332000", "206860683821114948625", "5968372055889205462500" ]

Expansion of e.g.f. exp(x^4/(24 * (1 - x)^4)).

A361578

[ "1", "0", "1", "1", "5", "8", "30", "85", "382", "1550", "7352" ]

Number of 5-connected polyhedra (or 5-connected simple planar graphs) with n nodes

A361579

[ "1", "0", "1", "0", "3", "1", "0", "51", "12", "1", "0", "3614", "447", "34", "1", "0", "991930", "53675", "2885", "85", "1", "0", "1051469032", "21514470", "741455", "16665", "201", "1", "0", "4366988803688", "30405612790", "642187105", "9816380", "90678", "462", "1", "0", "71895397383029040", "160152273169644", "2024633081100", "19625842425", "122330544", "474138", "1044", "1" ]

Triangular array read by rows. T(n,k) is the number of labeled digraphs on [n] with exactly k source-like components, n >= 0, 0 <= k <= n.

A361580

[ "1", "2", "3", "2", "5", "32", "7", "42", "3", "52", "11", "6432", "13", "72", "53", "842", "17", "9632", "19", "10542", "73", "112", "23", "1286432", "5", "132", "93", "14742", "29", "15106532", "31", "16842", "113", "172", "75", "181296432", "37", "192", "133", "20108542", "41", "21147632", "43", "221142", "15953", "232", "47", "24161286432", "7", "251052" ]

If n is composite, replace n with the concatenation of its nontrivial divisors, written in decreasing order, each divisor being written in base 10 with its digits in normal order, otherwise a(n) = n.

A361581

[ "1", "2", "3", "2", "5", "32", "7", "42", "3", "52", "11", "6432", "13", "72", "53", "842", "17", "9632", "19", "1542", "73", "112", "23", "2186432", "5", "312", "93", "41742", "29", "51016532", "31", "61842", "113", "712", "75", "812196432", "37", "912", "313", "2018542", "41", "12417632", "43", "221142", "51953", "322", "47", "42612186432", "7", "520152" ]

If n is composite, replace n with the concatenation of its nontrivial divisors, written in decreasing order, each divisor being written in base 10 with its digits in reverse order, otherwise a(n) = n.

A361582

[ "1", "0", "1", "0", "1", "2", "0", "5", "5", "6", "0", "83", "62", "42", "31", "0", "5048", "2494", "1172", "592", "302", "0", "1047008", "330063", "103961", "38312", "15616", "5984", "0", "705422362", "137934757", "28095923", "7243110", "2297690", "795930", "243668", "0", "1580348371788", "184557780045", "23226116293", "3951426731", "914429926", "261269562", "79512478", "20286025" ]

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

A361583

[ "1", "1", "3", "12", "88", "1217", "34672", "2039085", "246005109", "60296886108", "29828186693218", "29663937774464786", "59172529527454608139", "236453014376786629601848", "1891427400988740573006253862", "30274661556583530830890359188257", "969429810937979825934973090455224882" ]

Number of digraphs on n unlabeled nodes whose strongly connected components are complete digraphs.

A361584

[ "1", "1", "3", "12", "88", "1239", "36540", "2226595", "277421616", "69974281748", "35535207035048", "36224521019293188", "74004483908461354689", "302712665772844097945072", "2477999475270966827490305948", "40583406022745170376459610683073", "1329552679157905406495248763876363056" ]

Number of digraphs on n unlabeled nodes whose strongly connected components are directed cycles or single vertices.

A361585

[ "1", "0", "1", "1", "8", "28", "736", "17879", "1568614", "196581247", "62857465075", "34431266945361", "42146672798547398", "95881304594606248756", "459546334152150732106700", "4253461062245855670436620669", "80700118619568448244440535541825", "3011390106783578987361705575335328331" ]

Number of digraphs on n unlabeled nodes whose strongly connected components are directed cycles.

A361586

[ "1", "0", "1", "5", "90", "5289", "1071691", "712342075", "1585944117738", "12152982231404393", "328276896613548366675", "31834464336872565979301363", "11234630426387288679040317490771", "14576388456695908232721134339830232699", "70075904005979773819582865772534172929477101" ]

Number of directed graphs on n unlabeled nodes in which every node belongs to a directed cycle.

A361587

[ "1", "0", "1", "0", "1", "1", "0", "5", "4", "4", "0", "83", "56", "36", "24", "0", "5048", "2406", "1101", "542", "267", "0", "1047008", "324917", "101307", "37017", "14947", "5647", "0", "705422362", "136882286", "27757789", "7134897", "2257234", "779257", "237317", "0", "1580348371788", "183851281949", "23086772643", "3922864504", "907027520", "258909828", "78691767", "20035307" ]

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

A361588

[ "1", "0", "0", "0", "1", "1", "0", "5", "4", "4", "0", "83", "57", "37", "25", "0", "5048", "2411", "1110", "550", "271", "0", "1047008", "325015", "101467", "37140", "15024", "5682", "0", "705422362", "136887749", "27765860", "7139149", "2259378", "780314", "237684", "0", "1580348371788", "183852357683", "23088181536", "3923330808", "907186816", "258971872", "78716548", "20042357" ]

Triangle read by rows: T(n,k) is the number of digraphs on n unlabeled nodes with k strongly connected components and without isolated nodes.

A361589

[ "1", "0", "1", "4", "25", "271", "5682", "237684", "20042357", "3404651985", "1162523674892", "796395726736678", "1093229314594543016", "3004753338859186373234", "16527845763725396055765240", "181891586856152393087373330332", "4004313490358484085907684748704180", "176328671349936542115174881107633828418" ]

Number of acyclic digraphs on n unlabeled nodes without isolated nodes.

A361590

[ "1", "0", "1", "1", "0", "2", "5", "5", "0", "6", "90", "55", "42", "0", "31", "5289", "2451", "974", "592", "0", "302", "1071691", "323709", "94332", "29612", "15616", "0", "5984", "712342075", "135208025", "25734232", "6059018", "1650492", "795930", "0", "243668", "1585944117738", "181427072519", "21650983294", "3358042412", "704602272", "174576110", "79512478", "0", "20286025" ]

Triangle read by rows: T(n,k) is the number of digraphs on n unlabeled nodes with exactly k strongly connected components of size 1.

A361592

[ "1", "0", "1", "1", "0", "3", "18", "21", "0", "25", "1699", "1080", "774", "0", "543", "587940", "267665", "103860", "59830", "0", "29281", "750744901", "225144360", "64169325", "19791000", "10110735", "0", "3781503", "3556390155318", "672637205149", "126726655860", "29445913175", "7939815030", "3767987307", "0", "1138779265" ]

Triangular array read by rows. T(n,k) is the number of labeled digraphs on [n] with exactly k strongly connected components of size 1, n>=0, 0<=k<=n.