nyuuzyou/wikis huggingface dataset:
27,418, 521,963, 671,501, 273,905, 190,135, 082,692, 041,730, 405,303, 870,249, 023,209

== Number ==
$27 \, 418 \, 521 \, 963 \, 671 \, 501 \, 273 \, 905 \, 190 \, 135 \, 082 \, 692 \, 041 \, 730 \, 405 \, 303 \, 870 \, 249 \, 023 \, 209$ is:
:$3^9 \times 7^3 \times 11^3 \times 13^3 \times 17^3 \times 41^3 \times 43^3 \times 47^3 \times 443^3 \times 499^3 \times 3583^3$
:The $30 \, 154 \, 214 \, 043 \, 975 \, 990 \, 969$th cube number:
::$27 \, 418 \, 521 \, 963 \, 671 \, 501 \, 273 \, 905 \, 190 \, 135 \, 082 \, 692 \, 041 \, 730 \, 405 \, 303 \, 870 \, 249 \, 023 \, 209 = 30 \, 154 \, 214 \, 043 \, 975 \, 990 \, 969^3$
:The smallest cube $N$ whose such that the divisor sum of $N$ is also a cube:
== Also see ==
* Smallest Cube whose Sum of Divisors is Cube
* 
== Sources ==
* : $3^9 7^3 11^3 13^3 17^3 41^3 43^3 47^3 443^3 499^3 3583^3$
Category:Specific Numbers
Category:27,418, 521,963, 671,501, 273,905, 190,135, 082,692, 041,730, 405,303, 870,249, 023,209
Category:Integers whose Divisor Sum is Cube
27,594

== Number ==
$27 \, 594$ (twenty-seven thousand, five hundred and ninety-four) is:
:$2 \times 3^3 \times 7 \times 73$
:Can be expressed as a product in the following $2$ related ways:
::$27 \, 594 = 73 \times 9 \times 42 = 7 \times 3942$
== Sources ==
* 
* : $27,594$
* : $27,594$
Category:Specific Numbers
Category:27,594
27/Historical Note

== Historical Note on $27$ ==
The total score of all the coloured balls in is $27$:
and $2 + 3 + 4 + 5 + 6 + 7 = 27$, which is one less than the $7$th triangular number $28$.
== Sources ==
* : $15$
* : $27$
* : $27$
Category:Historical Notes
270

== Number ==
$270$ (two hundred and seventy) is:
:$2 \times 3^3 \times 5$
:The $5$th Ore number after $1$, $6$, $28$, $140$:
::$\dfrac {270 \times \map {\sigma_0} {270} } {\map {\sigma_1} {270} } = 6$
:and the $4$th after $1, 6, 140$ whose divisors also have an arithmetic mean which is an integer:
::$\dfrac {\map {\sigma_1} {270} } {\map {\sigma_0} {270} } = 45$
:The $17$th (and last) of the $17$ positive integers for which the value of the Euler $\phi$ function is $72$:
::$73$, $91$, $95$, $111$, $117$, $135$, $146$, $148$, $152$, $182$, $190$, $216$, $222$, $228$, $234$, $252$, $270$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:270
270,343

== Number ==
$270 \, 343$ (two hundred and seventy thousand, three hundred and forty-three) is:
:The $23 \, 674$th prime number
:The index of the $9$th, and largest known, (probable) repunit prime after $R_2$, $R_{19}$, $R_{23}$, $R_{317}$, $R_{1031}$, $R_{49 \, 081}$, $R_{86 \, 453}$, $R_{109 \, 297}$
== Also see ==
* 
* 
Category:Specific Numbers
Category:270,343
2701

== Number ==
$2701$ (two thousand, seven hundred and one) is:
:$37 \times 73$
:The $10$th Poulet number after $341$, $561$, $645$, $1105$, $1387$, $1729$, $1905$, $2047$, $2465$:
::$2^{2701} \equiv 2 \pmod {2701}$: $2701 = 37 \times 73$
:The $13$th Fermat pseudoprime to base $3$ after $91$, $121$, $286$, $671$, $703$, $949$, $1105$, $1541$, $1729$, $1891$, $2465$, $2665$:
::$3^{2701} \equiv 3 \pmod {2701}$
:The $37$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $1431$, $1540$, $1653$, $1770$, $1891$, $2016$, $2145$, $2278$, $2415$, $2556$:
::$2701 = \ds \sum_{k \mathop = 1}^{37} \paren {4 k - 3} = 37 \paren {2 \times 37 - 1}$
:The $73$rd triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $2211$, $2278$, $2346$, $2415$, $2485$, $2556$, $2628$:
::$2701 = \ds \sum_{k \mathop = 1}^{73} k = \dfrac {73 \times \paren {73 + 1} } 2$
:Hence $2701 = 37 \times 73$ is both the $37$th hexagonal number, and the $73$rd triangular number.
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:2701
2704

== Number ==
$2704$ (two thousand, seven hundred and four) is:
:$2^4 \times 13^2$
:The $19$th square number after $1$, $4$, $36$, $121$, $144$, $256$, $324$, $400$, $576$, $784$, $900$, $961$, $1024$, $1296$, $1600$, $1764$, $1936$, $2304$ to be the divisor sum value of some (strictly) positive integer:
::$2704 = \map {\sigma_1} {2359}$
:The $52$nd square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $1521$, $1600$, $1681$, $1764$, $1849$, $1936$, $2025$, $2116$, $2209$, $2304$, $2401$, $2500$, $2601$:
::$2704 = 52 \times 52$
== Also see ==
* 
* 
Category:Specific Numbers
Category:2704
271

== Number ==
$271$ (two hundred and seventy-one) is:
:The $58$th prime number
:The $10$th centered hexagonal number after $1$, $7$, $19$, $37$, $61$, $91$, $127$, $169$, $217$:
::$271 = 1 + 6 + 12 + 18 + 24 + 30 + 36 + 42 + 48 + 54 = 10^3 - 9^3$
:The $12$th of $29$ primes of the form $2 x^2 + 29$:
::$2 \times 11^2 + 29 = 271$ ()
:The larger of the $18$th pair of twin primes, with $269$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:271
272

== Number ==
$272$ (two hundred and seventy-two) is:
:$2^4 \times 17$
:The $5$th primitive abundant number after $20, 70, 88, 104$:
::$1 + 2 + 4 + 8 + 16 + 17 + 34 + 68 + 136 = 286 > 272$
:The $6$th primitive semiperfect number after $6, 20, 28, 88, 104$:
::$272 = 1 + 16 + 17 + 34 + 68 + 136$
:The $13$th inconsummate number after $62$, $63$, $65$, $75$, $84$, $95$, $161$, $173$, $195$, $216$, $261$, $266$:
::$\nexists n \in \Z_{>0}: n = 272 \times \map {s_{10} } n$
:The $53$rd positive integer $n$ such that no factorial of an integer can end with $n$ zeroes.
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:272
272,400,600

== Number ==
$272 \, 400 \, 600$ is:
:$2^3 \times 3 \times 5^2 \times 107 \times 4243$
:The index of the $1$st harmonic number to be greater than $20$:
::$H_{272 \, 400 \, 600} = \ds \sum_{k \mathop = 1}^{272 \, 400 \, 600} \frac 1 k \approx 20 \cdotp 00000 \, 00016$
== Also see ==
* First Harmonic Number to exceed 20
== Sources ==
* : $272,400,600$
* : $272,400,600$
Category:Specific Numbers
Category:272,400,600
2728

== Number ==
$2728$ (two thousand, seven hundred and twenty-eight) is:
:$2^3 \times 11 \times 31$
:The $10$th Kaprekar number after $1$, $9$, $45$, $55$, $99$, $297$, $703$, $999$, $2223$:
::$2728^2 = 7 \, 441 \, 984 \to 744 + 1984 = 2728$
== Also see ==
* 
Category:Specific Numbers
Category:2728
273

== Number ==
$273$ (two hundred and seventy-three) is:
:$3 \times 7 \times 13$
:The $28$th sphenic number after $30$, $42$, $66$, $70$, $\ldots$, $182$, $186$, $190$, $195$, $222$, $230$, $231$, $246$, $255$, $258$, $266$:
::$273 = 3 \times 7 \times 13$
:The $51$st lucky number:
::$1$, $3$, $7$, $9$, $13$, $15$, $21$, $\ldots$, $219$, $223$, $231$, $235$, $237$, $241$, $259$, $261$, $267$, $273$, $\ldots$
:The $3$rd of $4$ integers $n$ such that $n + 2 n$ can be expressed as a sum using each of the digits $1$ to $9$ exactly once each:
::$273 + 546 = 819$
:The product with its reversal equals the product of another $3$-digit number with its reversal:
::$273 \times 372 = 156 \times 651$
== Also see ==
* 
* 
* 
* 
== Sources ==
* : $192$
Category:Specific Numbers
Category:273
274

== Number ==
$274$ (two hundred and seventy-four) is:
:$2 \times 137$
:The $11$th Smith number after $4$, $22$, $27$, $58$, $85$, $94$, $121$, $166$, $202$, $265$:
::$2 + 7 + 4 = 2 + 1 + 3 + 7 = 13$
:The $27$th noncototient after $10$, $26$, $34$, $50$, $\ldots$, $244$, $260$, $266$, $268$:
::$\nexists m \in \Z_{>0}: m - \map \phi m = 274$
:where $\map \phi m$ denotes the Euler $\phi$ function
:The $43$rd nontotient:
::$\nexists m \in \Z_{>0}: \map \phi m = 274$
:where $\map \phi m$ denotes the Euler $\phi$ function
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:274
274,177

== Number ==
$274 \, 177$ is:
:The $23 \, 974$th prime.
:One of the two prime factors of the $6$th Fermat number $2^{\paren {2^6} } + 1$, the other being $67 \, 280 \, 421 \, 310 \, 721$:
::$274 \, 177 = 1071 \times 2^8 + 1$
== Also see ==
* 
* 
== Sources ==
* : $257$
* : $257$
Category:Specific Numbers
Category:274,177
274,924

== Number ==
$274 \, 924$ (two hundred and seventy-four thousand, nine hundred and twenty-four) is:
:$2^2 \times 13 \times 17 \times 311$
:The $14$th term of a sociable chain of order $28$ whose smallest term is $14 \, 316$:
::$14 \, 316$, $19 \, 116$, $31 \, 704$, $47 \, 616$, $83 \, 328$, $177 \, 792$, $295 \, 488$, $629 \, 072$, $589 \, 786$, $294 \, 896$, $358 \, 336$, $418 \, 904$, $366 \, 556$, $274 \, 924$, $275 \, 444$, $243 \, 760$, $376 \, 736$, $381 \, 028$, $285 \, 778$, $152\, 990$, $122 \, 410$, $97 \, 946$, $48 \, 976$, $45 \, 946$, $22 \, 976$, $22 \, 744$, $19 \, 916$, $17 \, 716$, $14 \, 316$, $\ldots$
== Also see ==
* 
Category:Specific Numbers
Category:274,924
2744

== Number ==
$2744$ (two thousand, seven hundred and forty-four) is:
:$2^3 \times 7^3$
:The $3$rd power of $14$ after $(1)$, $14$, $196$:
::$2744 = 14^3$
:The $14$th cube number after $1$, $8$, $27$, $64$, $125$, $216$, $343$, $512$, $729$, $1000$, $1331$, $1728$, $2197$:
::$2744 = 14 \times 14 \times 14$
== Also see ==
* 
* 
Category:Specific Numbers
Category:2744
275,305,224

== Number ==
$275 \, 305 \, 224$ is:
:$2^3 \times 3 \times 109 \times 105 \, 239$
:Up to rotations and reflections, the number of distinct magic squares of order $5$
== Also see ==
* Number of Magic Squares of Order 5
== Sources ==
* : $275,305,224$
* : $275,305,224$
Category:Specific Numbers
Category:275,305,224
275,444

== Number ==
$275 \, 444$ (two hundred and seventy-five thousand, four hundred and forty-four) is:
:$2^2 \times 13 \times 5297$
:The $15$th term of a sociable chain of order $28$ whose smallest term is $14 \, 316$:
::$14 \, 316$, $19 \, 116$, $31 \, 704$, $47 \, 616$, $83 \, 328$, $177 \, 792$, $295 \, 488$, $629 \, 072$, $589 \, 786$, $294 \, 896$, $358 \, 336$, $418 \, 904$, $366 \, 556$, $274 \, 924$, $275 \, 444$, $243 \, 760$, $376 \, 736$, $381 \, 028$, $285 \, 778$, $152\, 990$, $122 \, 410$, $97 \, 946$, $48 \, 976$, $45 \, 946$, $22 \, 976$, $22 \, 744$, $19 \, 916$, $17 \, 716$, $14 \, 316$, $\ldots$
== Also see ==
* 
Category:Specific Numbers
Category:275,444
2752

== Number ==
$2752$ (two thousand, seven hundred and fifty-two) is:
:$2^6 \times 43$
:The $43$rd pentagonal number after $1$, $5$, $12$, $22$, $35$, $\ldots$, $1820$, $1926$, $2035$, $2147$, $2262$, $2380$, $2501$, $2625$:
::$2752 = \ds \sum_{k \mathop = 1}^{43} \paren {3 k - 2} = \dfrac {43 \paren {3 \times 43 - 1} } 2$
:The $85$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $2380$, $2420$, $2501$, $2542$, $2625$, $2667$:
::$2752 = \ds \sum_{k \mathop = 1}^{43} \paren {3 k - 2} = \dfrac {43 \paren {3 \times 43 - 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:2752
276

== Number ==
$276$ (two hundred and seventy-six) is:
:$2^2 \times 3 \times 23$
:The $1$st positive integer which begins an aliquot sequence whose final status is unknown.
:The $12$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $120$, $153$, $190$, $231$:
::$\ds 276 = \sum_{k \mathop = 1}^{12} \paren {4 k - 3} = 12 \left({2 \times 12 - 1}\right)$
:The $14$th inconsummate number after $62$, $63$, $65$, $75$, $84$, $95$, $161$, $173$, $195$, $216$, $261$, $266$, $272$:
::$\nexists n \in \Z_{>0}: n = 276 \times \map {s_{10} } n$
:The $19$th untouchable number after $2$, $5$, $52$, $88$, $96$, $120$, $124$, $146$, $162$, $188$, $206$, $210$, $216$, $238$, $246$, $248$, $262$, $268$
:The $23$rd triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $136$, $153$, $171$, $190$, $210$, $231$, $253$:
::$\ds 276 = \sum_{k \mathop = 1}^{23} k = \dfrac {23 \times \paren {23 + 1} } 2$
== Also see ==
* Aliquot Sequence starting from $276$
* 
* 
* 
* 
== Sources ==
* : $276$
* : $276$
Category:Specific Numbers
Category:276
277

== Number ==
$277$ (two hundred and seventy-seven) is:
:The $59$th prime number
:The $13$th near-repdigit prime after $101$, $113$, $131$, $151$, $181$, $191$, $199$, $211$, $223$, $227$, $229$, $233$
== Sources ==
* 
* 
Category:Specific Numbers
Category:277
277,777,788,888,899

== Number ==
$277 \, 777 \, 788 \, 888 \, 899$ is:
:$13 \times 59 \times 1699 \times 213 \, 161 \, 503$
:The smallest number having a multiplicative persistence of $11$.
== Also see ==
* 
== Sources ==
* : $277,777,788,888,899$
Category:Specific Numbers
Category:277,777,788,888,899
2775

== Number ==
$2775$ (two thousand, seven hundred and seventy-five) is:
:$3 \times 5^2 \times 37$
:The $74$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $2211$, $2278$, $2346$, $2415$, $2485$, $2556$, $2628$, $2701$:
::$2775 = \ds \sum_{k \mathop = 1}^{74} k = \dfrac {74 \times \paren {74 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:2775
278

== Number ==
$278$ (two hundred and seventy-eight) is:
:$2 \times 139$
:The $44$th nontotient:
::$\nexists m \in \Z_{>0}: \map \phi m = 278$
:where $\map \phi m$ denotes the Euler $\phi$ function
:The $54$th positive integer $n$ such that no factorial of an integer can end with $n$ zeroes.
== Also see ==
* 
* 
Category:Specific Numbers
Category:278
278,886

== Number ==
$278 \, 886$ (two hundred and seventy-eight thousand, eight hundred and eighty-six) is:
:$2 \times 3 \times 53 \times 877$
:Has a square which starts with a remarkably long string of $7$s:
::$278 \, 886^2 = 77 \, 777 \, 400 \, 996$
== Sources ==
* : $278,886$
* : $278,886$
Category:Specific Numbers
Category:278,886
279

== Number ==
$279$ (two hundred and seventy-nine) is:
:$3^2 \times 31$
:The $55$th positive integer $n$ such that no factorial of an integer can end with $n$ zeroes.
:Every positive integer can be expressed as the sum of at most $279$ positive $8$th powers.
== Also see ==
* 
* 
Category:Specific Numbers
Category:279
279,936

== Number ==
$279 \, 936$ (two hundred and seventy-nine thousand, nine hundred and thirty-six) is:
:$2^7 \times 3^7$
:The $6$th seventh power after $1$, $128$, $2187$, $16 \, 384$, $78 \, 125$:
::$279 \, 936 = 6 \times 6 \times 6 \times 6 \times 6 \times 6 \times 6$
:The $7$th power of $6$ after $(1)$, $6$, $36$, $216$, $1296$, $7776$, $46 \, 656$:
::$279 \, 936 = 6^7$
== Also see ==
* 
* 
Category:Specific Numbers
Category:279,936
2791

== Number ==
$2791$ (two thousand, seven hundred and ninety-one) is:
:The $406$th prime number
:The $31$st centered hexagonal number after $1$, $7$, $19$, $37$, $61$, $91$, $127$, $\ldots$, $1387$, $1519$, $1657$, $1801$, $1951$, $2107$, $2269$, $2437$, $2611$:
::$2791 = \ds 1 + \sum_{k \mathop = 1}^{31 - 1} 6 k = 31^3 - 30^3$
== Also see ==
* 
* 
Category:Specific Numbers
Category:2791
2795

== Number ==
$2795$ (two thousand, seven hundred and ninety-five) is:
:$5 \times 13 \times 43$
:The $43$rd second pentagonal number after $2$, $7$, $15$, $26$, $\ldots$, $1962$, $2072$, $2185$, $2301$, $2420$, $2542$, $2667$:
::$2795 = \dfrac {43 \paren {3 \times 43 + 1} } 2$
:The $86$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $2380$, $2420$, $2501$, $2542$, $2625$, $2667$, $2752$:
::$2795 = \dfrac {43 \paren {3 \times 43 + 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:2795
27 is Smallest Number whose Period of Reciprocal is 3

== Theorem ==
$27$ is the smallest positive integer the decimal expansion of whose reciprocal has a period of $3$:
:$\dfrac 1 {27} = 0 \cdotp \dot 03 \dot 7$
== Proof ==
From Reciprocal of $27$:
It can be determined by inspection of all smaller integers that this is indeed the smallest to have a period of $3$.
== Sources ==
* : $27$
* : $999$
* : $27$
* : $999$
Category:27
Category:Examples of Reciprocals
28

== Number ==
$28$ (twenty-eight) is:
:$2^2 \times 7$
:The only perfect number which is the sum of equal powers of exactly $2$ positive integers:
::$28 = 1^3 + 3^3$
:The $1$st triangular number which is the sum of $2$ cubes:
::$28 = 1 + 27 = 1^3 + 3^3$
:The $2$nd perfect number after $6$:
::$28 = 1 + 2 + 4 + 7 + 14 = \map {\sigma_1} {28} - 28 = 4 \times 7 = 2^{3 - 1} \paren {2^3 - 1}$
:The $3$rd Ore number after $1$, $6$:
::$\dfrac {28 \times \map {\sigma_0} {28} } {\map {\sigma_1} {28} } = 3$
:The $3$rd Keith number after $14$, $19$:
::$2$, $8$, $10$, $18$, $28$, $\ldots$
:The $3$rd primitive semiperfect number after $6$, $20$:
::$28 = 1 + 2 + 4 + 7 + 14$
:The $3$rd Sierpiński number of the first kind after $2$, $5$:
::$28 = 3^3 + 1$
:The $4$th hexagonal number after $1$, $6$, $15$:
::$28 = 1 + 5 + 9 + 13 = 4 \paren {2 \times 4 - 1}$
:The $5$th term of Göbel's sequence after $1$, $2$, $3$, $5$, $10$:
::$28 = \paren {1 + 1^2 + 2^2 + 3^2 + 5^2 + 10^2} / 5$
:The $6$th semiperfect number after $6$, $12$, $18$, $20$, $24$:
::$28 = 1 + 2 + 4 + 7 + 14$
:The $7$th triangular number after $1$, $3$, $6$, $10$, $15$, $21$:
::$28 = 1 + 2 + 3 + 4 + 5 + 6 + 7 = \dfrac {7 \times \paren {7 + 1} } 2$
:Hence there are $28$ dominoes in a standard set
:The $7$th happy number after $1$, $7$, $10$, $13$, $19$, $23$:
::$28 \to 2^2 + 8^2 = 4 + 64 = 68 \to 6^2 + 8^2 = 36 + 64 = 100 \to 1^2 + 0^2 + 0^2 = 1$
:The $12$th Ulam number after $1$, $2$, $3$, $4$, $6$, $8$, $11$, $13$, $16$, $18$, $26$:
::$28 = 2 + 26$
:The $12$th even number after $2$, $4$, $6$, $8$, $10$, $12$, $14$, $16$, $20$, $22$, $26$ which cannot be expressed as the sum of $2$ composite odd numbers.
:The $15$th positive integer which is not the sum of $1$ or more distinct squares:
::$2$, $3$, $6$, $7$, $8$, $11$, $12$, $15$, $18$, $19$, $22$, $23$, $24$, $27$, $28$, $\ldots$
:The $17$th after $1$, $2$, $4$, $5$, $6$, $8$, $9$, $12$, $13$, $15$, $16$, $17$, $20$, $24$, $25$, $27$ of the $24$ positive integers which cannot be expressed as the sum of distinct non-pythagorean primes.
:The $20$th integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $13$, $14$, $15$, $16$, $18$, $19$, $24$, $25$, $27$ such that $2^n$ contains no zero in its decimal representation:
::$2^{28} = 268 \, 435 \, 456$
== Also see ==
* Perfect Number which is Sum of Equal Powers of Two Numbers
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
== Historical Note ==
== Sources ==
* : $28$
* : $28$
Category:Specific Numbers
Category:28
28,561

== Number ==
$28 \, 561$ (twenty-eight thousand, five hundred and sixty-one) is:
:$13^4$
:The $4$th power of $13$ after $(1)$, $13$, $169$, $2197$:
::$28 \, 561 = 13^4$
:Can be expressed as the sum of $6$ fourth powers:
::$28 \, 561 = 13^4 = 12^4 + 8^4 + 7^4 + 6^4 + 2^4 + 2^4$
:The $13$th fourth power after $1$, $16$, $81$, $256$, $625$, $1296$, $2401$, $4096$, $6561$, $10 \, 000$, $14 \, 641$, $20 \, 736$:
::$28 \, 561 = 13 \times 13 \times 13 \times 13$
:The $169$th square number:
::$28 \, 561 = 169 \times 169$
== Also see ==
* Fourth Power expressible as Sum of 6 Fourth Powers
* 
* 
* 
== Sources ==
* : $28,561$
Category:Specific Numbers
Category:28,561
28/Historical Note

== Historical Note on $28$ ==
There are approximately $28$ days in the lunar cycle.
This is probably the origin of the widespread cultural custom of splitting time both into $7$-day chunks called weeks, and also the year into $12$ approximately $28$-day chunks called months.
There are $28$ pounds avoirdupois in one quarter.
== Sources ==
* : $28$
* : $28$
Category:Historical Notes
280

== Number ==
$280$ (two hundred and eighty) is:
:$2^3 \times 5 \times 7$
:The $10$th octagonal number, after $1$, $8$, $21$, $40$, $65$, $96$, $133$, $176$, $225$:
::$280 = 1 + 7 + 13 + 19 + 25 + 31 + 37 + 43 + 49 + 55 = 10 \paren {3 \times 10 - 2}$
:The $42$nd happy number after $1$, $7$, $10$, $13$, $19$, $23$, $\ldots$, $188$, $190$, $192$, $193$, $203$, $208$, $219$, $226$, $230$, $236$, $239$, $262$, $263$:
::$280 \to 2^2 + 8^2 + 0^2 = 4 + 64 + 0 = 68 \to 6^2 + 8^2 = 36 + 64 = 100 \to 1^2 + 0^2 + 0^2 = 1$
== Also see ==
* 
* 
Category:Specific Numbers
Category:280
2809

== Number ==
$2809$ (two thousand, eight hundred and nine) is:
:$53^2$
:The $53$rd square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $1521$, $1600$, $1681$, $1764$, $1849$, $1936$, $2025$, $2116$, $2209$, $2304$, $2401$, $2500$, $2601$, $2704$:
::$2809 = 53 \times 53$
:The $44$th positive integer which cannot be expressed as the sum of a square and a prime:
::$1$, $10$, $25$, $34$, $58$, $64$, $85$, $\ldots$, $1521$, $1681$, $1849$, $1906$, $1936$, $2116$, $2209$, $2304$, $2500$, $2809$, $\ldots$
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Category:Specific Numbers
Category:2809
281

== Number ==
$281$ (two hundred and eighty-one) is:
:The $60$th prime number
:The $1$st of a triplet of consecutive integers which have the property that if their digits are multiplied, and the process repeated on the result until only $1$ digit remains, that final digit is the same for all three:
::$2 \times 8 \times 1 = 16$; $1 \times 6 = 6$
:The smaller of the $19$th pair of twin primes, with $283$
:The $19$th Sophie Germain prime after $2$, $3$, $5$, $11$, $23$, $29$, $41$, $53$, $83$, $89$, $113$, $131$, $173$, $179$, $191$, $233$, $239$, $251$:
::$2 \times 281 + 1 = 563$, which is prime.
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* : $281$
Category:Specific Numbers
Category:281
282

== Number ==
$282$ (two hundred and eighty-two) is:
:$2 \times 3 \times 47$
:The $2$nd of a triplet of consecutive integers which have the property that if their digits are multiplied, and the process repeated on the result until only $1$ digit remains, that final digit is the same for all three:
::$2 \times 8 \times 2 = 32$; $3 \times 2 = 6$
:The $15$th number after $1$, $3$, $22$, $66$, $70$, $81$, $94$, $115$, $119$, $170$, $210$, $214$, $217$, $265$ whose divisor sum is square:
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:The $29$th sphenic number after $30$, $42$, $66$, $70$, $\ldots$, $182$, $186$, $190$, $195$, $222$, $230$, $231$, $246$, $255$, $258$, $266$, $273$:
::$282 = 2 \times 3 \times 47$
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== Sources ==
* : $281$
Category:Specific Numbers
Category:282
2821

== Number ==
$2821$ (two thousand, eight hundred and twenty-one) is:
:$7 \times 13 \times 31$
:The $5$th Carmichael number after $561$, $1105$, $1729$, $2465$:
::$\forall a \in \Z: a \perp 2821: a^{2820} \equiv 1 \pmod {2821}$
:The $9$th Fermat pseudoprime to base $5$ after $4$, $124$, $217$, $561$, $781$, $1541$, $1729$, $1891$:
::$5^{2821} \equiv 5 \pmod {2821}$
:The $11$th Poulet number after $341$, $561$, $645$, $1105$, $1387$, $1729$, $1905$, $2047$, $2465$, $2701$:
::$2^{2821} \equiv 2 \pmod {2821}$: $2821 = 7 \times 13 \times 31$
:The $14$th Fermat pseudoprime to base $3$ after $91$, $121$, $286$, $671$, $703$, $949$, $1105$, $1541$, $1729$, $1891$, $2465$, $2665$, $2701$:
::$3^{2821} \equiv 3 \pmod {2821}$
:The $31$st octagonal number, after $1$, $8$, $21$, $40$, $65$, $\ldots$, $1281$, $1408$, $1541$, $1680$, $1825$, $1976$, $2133$, $2296$, $2465$, $2640$:
::$2821 = \ds \sum_{k \mathop = 1}^{31} \paren {6 k - 5} = 31 \paren {3 \times 31 - 2}$
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== Sources ==
* : $2821$
Category:Specific Numbers
Category:2821
2829

== Number ==
$2829$ (two thousand, eight hundred and twenty-nine) is:
:$3 \times 23 \times 41$
:The $5$th positive integer after $353$, $651$, $2487$, $2501$ such that its fourth power is the sum of $4$ other fourth powers of positive integers with no common factors:
::$2829^4 = 850^4 + 1010^4 + 1546^4 + 2745^4$
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Category:Specific Numbers
Category:2829
283

== Number ==
$283$ (two hundred and eighty-three) is:
:The $61$st prime number
:The $3$rd of a triplet of consecutive integers which have the property that if their digits are multiplied, and the process repeated on the result until only $1$ digit remains, that final digit is the same for all three:
::$2 \times 8 \times 3 = 48$; $48 \times 6 = 32$; $3 \times 2 = 6$
:The smaller of the $4$th pair of primes whose prime gap is $10$:
::$293 - 283 = 10$
:The larger of the $19$th pair of twin primes, with $281$
:The $22$nd left-truncatable prime after $2$, $3$, $5$, $7$, $13$, $17$, $23$, $37$, $43$, $47$, $53$, $67$, $73$, $83$, $97$, $113$, $137$, $167$, $173$, $197$, $223$
:The $52$nd lucky number:
::$1$, $3$, $7$, $9$, $13$, $15$, $21$, $\ldots$, $219$, $223$, $231$, $235$, $237$, $241$, $259$, $261$, $267$, $273$, $283$, $\ldots$
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* : $281$
Category:Specific Numbers
Category:283
2835

== Number ==
$2835$ (two thousand, eight hundred and thirty-five) is:
:$3^4 \times 5 \times 7$
:The $4$th odd abundant number after $945$, $1575$, $2205$:
::$\map {\sigma_1} {2835} - 2835 = 2973 > 2835$
== Also see ==
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Category:Specific Numbers
Category:2835