nyuuzyou/wikis huggingface dataset:
1289

== Number ==
$1289$ (one thousand, two hundred and eighty-nine) is:
:The $209$th prime number
:The smaller of the $44$th pair of twin primes:
::$1291 - 1289 = 2$
:The $45$th Sophie Germain prime after $2$, $3$, $5$, $11$, $23$, $29$, $41$, $\ldots$, $761$, $809$, $911$, $953$, $1013$, $1019$, $1031$, $1049$, $1103$, $1223$, $1229$:
::$2 \times 1289 + 1 = 2579$, which is prime.
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:1289
129

== Number ==
$129$ (one hundred and twenty-nine) is:
:$3 \times 43$
:The $27$th lucky number:
::$1$, $3$, $7$, $9$, $13$, $15$, $21$, $25$, $31$, $33$, $37$, $43$, $49$, $51$, $63$, $67$, $73$, $75$, $79$, $87$, $93$, $99$, $105$, $111$, $115$, $127$, $129$, $\ldots$
:The $23$rd happy number after $1$, $7$, $10$, $13$, $19$, $23$, $\ldots$, $91$, $94$, $97$, $100$, $103$, $109$:
::$129 \to 1^2 + 2^2 + 9^2 = 1 + 4 + 81 = 86 \to 8^2 + 6^2 = 64 + 36 = 100 \to 1^2 + 0^2 + 0^2 = 1$
:The $25$th positive integer $n$ such that no factorial of an integer can end with $n$ zeroes.
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:129
129,572,008

== Number ==
$129 \, 572 \, 008$ is:
:$2^3 \times 79 \times 205019$
:The $40$th Keith number:
::$1$, $2$, $9$, $5$, $7$, $2$, $0$, $0$, $8$, $34$, $\ldots$, $32 \, 456 \, 930$, $64 \, 849 \, 899$, $129 \, 572 \, 008$, $\ldots$
== Also see ==
* 
== Sources ==
* : $129,572,008$
Category:Specific Numbers
Category:129,572,008
1296

== Number ==
$1296$ (one thousand two hundred and ninety-six) is:
:$2^4 \times 3^4$
:The $4$th power of $6$ after $(1)$, $6$, $36$, $216$:
::$1296 = 6^4$
:The $6$th fourth power after $1$, $16$, $81$, $256$, $625$:
::$1296 = 6 \times 6 \times 6 \times 6$
:The sum of the first $8$ cubes:
::$1296 = 1^3 + 2^3 + 3^3 + 4^3 + 5^3 + 6^3 + 7^3 + 8^3$
:The $14$th square number after $1$, $4$, $36$, $121$, $144$, $256$, $324$, $400$, $576$, $784$, $900$, $961$, $1024$ to be the divisor sum value of some (strictly) positive integer:
::$1296 = \map {\sigma_1} {510}$
:The $36$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $625$, $676$, $729$, $784$, $841$, $900$, $961$, $1024$, $1089$, $1156$, $1225$:
::$1296 = 36 \times 36$
:The $45$th Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$, $12$, $\ldots$, $1111$, $1112$, $1113$, $1115$, $1116$, $1131$, $1176$, $1184$, $1197$, $1212$:
::$1296 = 12 \times 108 = 12 \times \paren {1 \times 2 \times 9 \times 6}$
== Also see ==
* 
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:1296
1297

== Number ==
$1297$ (one thousand, two hundred and ninety-seven) is:
:The $211$th prime number
:The $11$th prime number of the form $n^2 + 1$ after $2$, $5$, $17$, $37$, $101$, $197$, $257$, $401$, $577$, $677$:
::$1297 = 36^2 + 1$
== Also see ==
* 
* 
Category:Specific Numbers
Category:1297
12 Knights to Attack or Occupy All Squares on Chessboard

== Theorem ==
On a standard chessboard, a minimum of $12$ knights are needed to ensure all squares are either occupied or under attack.
== Proof ==
First, we show that fewer than $12$ knights are not enough to occupy or attack each square.
Consider the $12$ squares $\text a 1$, $\text a 2$, $\text b 2$, $\text a 8$, $\text b 8$, $\text b 7$, $\text h 8$, $\text h 7$, $\text g 7$, $\text h 1$, $\text g 1$ and $\text g 2$.
No knight can occupy or attack more than one of these squares.
Hence, fewer than $12$ knights are not sufficient.
Second, we show that $12$ knights can be placed to ensure all squares are either occupied or under attack.
For example, with the knights being on $\text b 3$, $\text c 3$, $\text c 4$, $\text c 7$, $\text c 6$, $\text d 6$, $\text g 6$, $\text f 6$, $\text f 5$, $\text f 2$, $\text f 3$ and $\text e 3$, all squares are either occupied or under attack:
== Sources ==
* : $12$
Category:12
Category:Recreational Chess
12 times Divisor Sum of 12 equals 14 times Divisor Sum of 14

== Theorem ==
$x = 12$ and $y = 14$ are solutions to the indeterminate equation:
:$x \, \map {\sigma_1} x = y \, \map {\sigma_1} y$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
== Sources ==
* : $12$
Category:12
Category:14
Category:Divisor Sum Function
13,033

== Number ==
$13 \, 033$ (thirteen thousand and thirty-three) is:
:The $1552$nd prime number
:The $17$th prime $p$ such that $p \# - 1$, where $p \#$ denotes primorial (product of all primes up to $p$) of $p$, is prime:
::$3$, $5$, $11$, $13$, $41$, $89$, $317$, $337$, $991$, $1873$, $2053$, $2377$, $4093$, $4297$, $4583$, $6569$, $13 \, 033$
== Also see ==
* 
* 
Category:Specific Numbers
Category:13,033
13,123,110

== Number ==
$13 \, 123 \, 110$ is:
:$2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 19 \times 23$
:The area of each of the smallest set of $3$ Pythagorean triangles which all have the same area:
::$4485-5852-7373$ triangle, $3059-8580-9109$ triangle, $1380-19 \, 019-19 \, 069$ triangle
== Also see ==
* Smallest Triplet of Primitive Pythagorean Triangles with Same Area
* 
== Sources ==
* : $13,123,110$
Category:Specific Numbers
Category:13,123,110
13,649

== Number ==
$13 \, 649$ (thirteen thousand, six hundred and forty-nine) is:
:The $1613$rd prime number
:The $15$th prime $p$ such that $p \# + 1$, where $p \#$ denotes primorial (product of all primes up to $p$) of $p$, is prime:
::$2$, $3$, $5$, $7$, $11$, $31$, $379$, $1019$, $1021$, $2657$, $3229$, $4547$, $4787$, $11 \, 549$, $13 \, 649$
== Also see ==
* 
* 
Category:Specific Numbers
Category:13,649
13,832

== Number ==
$13 \, 832$ (thirteen thousand, eight hundred and thirty-two) is:
:$2^3 \times 7 \times 13 \times 19$
:The $3$rd taxicab number after $1729$, $4104$: a positive integer which can be expressed as the sum of $2$ positive cubes in $2$ different ways:
::$13 \, 832 = 24^3 + 2^3 = 20^3 + 18^3$
== Also see ==
* 
Category:Specific Numbers
Category:13,832
13/Historical Note

== Historical Note on $13$ (Thirteen) ==
The number $13$ is traditionally unlucky.
This is apparently based on the fact that there were $13$ people who attended the Last Supper.
It is believed that this superstition originated in the middle ages.
The word triskaidekaphobia means fear of the number $13$.
The fact that $13$ is in fact classified by number theorists as a lucky number is just one more indication of how mathematicians delight in confusing muggles.
Apart from that, there are several instances of the number $13$ in the mundane world:
:There are $13$ times $4$ weeks in a year
:There are $13$ cards in each of the $4$ suits of a deck of cards
:The number of objects in a baker's dozen.
== Sources ==
* : $13$
* : $13$
::(although in this edition there is a misprint of triakaidekaphobia fpr triskaidekaphobia)
Category:Historical Notes
130

== Number ==
$130$ (one hundred and thirty) is:
:$2 \times 5 \times 13$
:The magic constant of a magic cube of order $4$, after $1$, $(9)$, $42$:
::$\ds 130 = \dfrac 1 {4^2} \sum_{k \mathop = 1}^{4^3} k = \dfrac {4 \paren {4^3 + 1} } 2$
:The $5$th positive integer after $50$, $65$, $85$, $125$ which can be expressed as the sum of two square numbers in two or more different ways:
::$130 = 11^2 + 3^2 = 9^2 + 7^2$
:The $6$th non-square positive integer which cannot be expressed as the sum of a square and a prime:
::$10$, $34$, $58$, $85$, $91$, $130$, $\ldots$
:The $10$th sphenic number after $30$, $42$, $66$, $70$, $78$, $102$, $105$, $110$, $114$:
::$130 = 2 \times 5 \times 13$
:The $10$th positive integer which cannot be expressed as the sum of a square and a prime:
::$1$, $10$, $25$, $34$, $58$, $64$, $85$, $91$, $121$, $130$, $\ldots$
:The $11$th noncototient after $10$, $26$, $34$, $50$, $52$, $58$, $86$, $100$, $116$, $122$:
::$\nexists m \in \Z_{>0}: m - \map \phi m = 130$
:where $\map \phi m$ denotes the Euler $\phi$ function
:The $11$th integer after $7$, $13$, $19$, $35$, $38$, $41$, $57$, $65$, $70$, $125$ the decimal representation of whose square can be split into two parts which are each themselves square:
::$130^2 = 16 \, 900$; $16 = 4^2$, $900 = 30^2$
:The $24$th happy number after $1$, $7$, $10$, $13$, $19$, $23$, $\ldots$, $91$, $94$, $97$, $100$, $103$, $109$, $129$:
::$130 \to 1^2 + 3^2 + 0^2 = 1 + 9 + 0 = 10 \to 1^2 + 0^2 = 1$
== Also see ==
* 
* 
* 
* 
* 
* 
* 
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Category:Specific Numbers
Category:130
1306

== Number ==
$1306$ (one thousand three hundred and six) is:
:$2 \times 653$
:The $5$th number after $1$, $9$, $20$, $180$ whose square has a divisor sum which is itself square:
:The $16$th number after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $89$, $135$, $175$, $518$, $598$ which is the sum of the increasing powers of its digits taken in order:
::$1^1 + 3^2 + 0^3 + 6^4 = 1306$
== Also see ==
* 
* 
Category:Specific Numbers
Category:1306
1307

== Number ==
$1307$ (one thousand, three hundred and seven) is:
:The $214$th prime number
:The $29$th safe prime after $5$, $7$, $11$, $23$, $47$, $\ldots$, $479$, $503$, $563$, $587$, $719$, $839$, $863$, $887$, $983$, $1019$, $1187$, $1283$:
::$1307 = 2 \times 653 + 1$
== Also see ==
* 
* 
Category:Specific Numbers
Category:1307
1309

== Number ==
$1309$ (one thousand, three hundred and nine) is:
:$7 \times 11 \times 17$
:The $1$st term of the $1$st triplet of consecutive positive integers all of which are sphenic:
::$1309 = 7 \times 11 \times 17$, $1310 = 2 \times 5 \times 131$, $1311 = 3 \times 19 \times 23$
== Also see ==
* 
== Sources ==
* : $1309$
Category:Specific Numbers
Category:1309
131

== Number ==
$131$ (one hundred and thirty-one) is:
:The $32$nd prime number
:The $2$nd positive integer after $61$ whose reciprocal, when expressed in decimal notation, contains an equal number ($13$) of each of the digits from $0$ to $9$:
::$\dfrac 1 {131} = 0 \cdotp \dot 00763 \, 35877 \, 86259 \, 54198 \, 47328 \, 24427 \, 48091 \, 60305 \, 34351 \, 14503 \, 81679 \, 38931 \, 29770 \, 99236 \, 64122 \, 13740 \, 45801 \, 52671 \, 75572 \, 51908 \, 39694 \, 65648 \, 85496 \, 18320 \, 61068 \, 7022 \dot 9$
:The $3$rd near-repdigit prime after $101$, $113$
:The $7$th palindromic prime after $2$, $3$, $5$, $7$, $11$, $101$
:The $12$th Sophie Germain prime after $2$, $3$, $5$, $11$, $23$, $29$, $41$, $53$, $83$, $89$, $113$:
::$2 \times 131 + 1 = 263$, which is prime
:The $12$th long period prime after $7$, $17$, $19$, $23$, $29$, $47$, $59$, $61$, $97$, $109$, $113$:
::$\dfrac 1 {131} = 0 \cdotp \dot 00763 \, 35877 \, 86259 \, 54198 \, 47328 \, 24427 \, 48091 \, 60305 \, 34351 \, 14503 \, 81679 \, 38931 \, 29770 \, 99236 \, 64122 \, 13740 \, 45801 \, 52671 \, 75572 \, 51908 \, 39694 \, 65648 \, 85496 \, 18320 \, 61068 \, 7022 \dot 9$
:The $15$th permutable prime after $2$, $3$, $5$, $7$, $11$, $13$, $17$, $31$, $37$, $71$, $73$, $79$, $97$, $113$
== Also see ==
* 
* 
* 
* 
* 
* 
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Category:Specific Numbers
Category:131
131,071

== Number ==
$131 \, 071$ (one hundred and thirty-one thousand and seventy-one) is:
:The $12 \, 251$st prime number
:The $6$th Mersenne prime after $3$, $7$, $31$, $127$, $8191$:
::$131 \, 071 = 2^{17} - 1$
:The $7$th Mersenne number after $3$, $7$, $31$, $127$, $2047$, $8191$:
::$131 \, 071 = 2^{17} - 1$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:131,071
131,072

== Number ==
$131 \, 072$ (one hundred and thirty-one thousand and seventy-two) is:
:$2^{17}$
:The $17$th power of $2$ after $(1)$, $2$, $4$, $8$, $16$, $32$, $64$, $128$, $256$, $512$, $1024$, $2048$, $4096$, $8192$, $16 \, 384$, $32 \, 768$, $65 \, 536$:
::$131 \, 072 = 2^{17}$
:The $18$th almost perfect number after $1$, $2$, $4$, $8$, $16$, $32$, $64$, $128$, $256$, $512$, $1024$, $2048$, $4096$, $8192$, $16 \, 384$, $32 \, 768$, $65 \, 536$:
::$\map {\sigma_1} {131 \, 072} = 262 \, 143 = 2 \times 131 \, 072 - 1$
== Also see ==
* 
* 
Category:Specific Numbers
Category:131,072
1310

== Number ==
$1310$ (one thousand, three hundred and ten) is:
:$2 \times 5 \times 131$
:The $2$nd term of the $1$st triplet of consecutive positive integers all of which are sphenic:
::$1309 = 7 \times 11 \times 17$, $1310 = 2 \times 5 \times 131$, $1311 = 3 \times 19 \times 23$
== Also see ==
* 
Category:Specific Numbers
Category:1310
1311

== Number ==
$1311$ (one thousand, three hundred and eleven) is:
:$3 \times 19 \times 23$
:The $3$rd term of the $1$st triplet of consecutive positive integers all of which are sphenic:
::$1309 = 7 \times 11 \times 17$, $1310 = 2 \times 5 \times 131$, $1311 = 3 \times 19 \times 23$
:The $46$th Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$, $12$, $\ldots$, $1111$, $1112$, $1113$, $1115$, $1116$, $1131$, $1176$, $1184$, $1197$, $1212$, $1296$:
::$1311 = 437 \times 3 = 437 \times \paren {1 \times 3 \times 1 \times 1}$
== Also see ==
* 
* 
Category:Specific Numbers
Category:1311
1312

== Number ==
$1312$ (one thousand, three hundred and twelve) is:
:$2^5 \times 41$
:The $16$th primitive abundant number after $20$, $70$, $88$, $104$, $272$, $304$, $368$, $464$, $550$, $572$, $650$, $748$, $836$, $945$, $1184$:
::$1 + 2 + 4 + 8 + 16 + 32 + 41 + 82 + 164 + 328 + 656 = 1334 > 1312$
:The $22$nd primitive semiperfect number after $6$, $20$, $28$, $88$, $104$, $\ldots$, $572$, $650$, $748$, $770$, $910$, $945$, $1184$, $1190$:
::$1312 = 1 + 8 + 32 + 41 + 82 + 164 + 328 + 656$
== Also see ==
* 
* 
Category:Specific Numbers
Category:1312
1319

== Number ==
$1319$ (one thousand, three hundred and nineteen) is:
:The $215$th prime number
:The $30$th safe prime after $5$, $7$, $11$, $23$, $47$, $\ldots$, $839$, $863$, $887$, $983$, $1019$, $1187$, $1283$, $1307$:
::$1319 = 2 \times 659 + 1$
:The smaller of the $46$th pair of twin primes:
::$1321 - 1319 = 2$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:1319
132

== Number ==
$132$ (one hundred and thirty-two) is:
:$2^2 \times 3 \times 11$
:The $1$st (positive) integer which is the sum of all the $2$-digit (positive) integers formed from its own digits:
::$132 = 12 + 13 + 21 + 23 + 31 + 32$
:The $6$th Catalan number after $(1)$, $1$, $2$, $5$, $14$, $42$:
::$132 = \dfrac 1 {6 + 1} \dbinom {2 \times 6} 6 = \dfrac 1 7 \times 924$
:The $19$th Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$, $12$, $15$, $24$, $36$, $111$, $112$, $115$, $128$:
::$132 = 22 \times 6 = 22 \times \left({1 \times 3 \times 2}\right)$
== Also see ==
* 132 is Sum of all 2-Digit Numbers formed from its Digits
* 
* 
== Sources ==
* : $132$
* : $132$
Category:Specific Numbers
Category:132
132,049

== Number ==
$132 \, 049$ (one hundred and thirty-two thousand and forty-nine) is:
:The $12 \, 331$st prime number
:The index of the $30$th Mersenne prime after $2$, $3$, $5$, $7$, $13$, $\ldots$, $11 \, 213$, $19 \, 937$, $21 \, 701$, $23 \, 209$, $44 \, 497$, $86 \, 243$, $110 \, 503$:
::$M_{132 \, 049} = 2^{132 \, 049} - 1 \approx 5 \cdotp 127 \times 10^{39 \, 750}$
== Also see ==
* 
* 
== Historical Note ==
== Sources ==
* : $2^{132049} - 1$
* : $2^{132,049} - 1$
Category:Specific Numbers
Category:132,049
132,049/Historical Note

== Historical Note on $132 \, 049$ ==
The Mersenne number $M_{132 \, 049} = 2^{132 \, 049} - 1$ was demonstrated to be a Mersenne prime on $19$ September $1983$ by , using two CRAY-1 supercomputers linked together.
== Sources ==
* : $2^{132049} - 1$
* : $2^{132,049} - 1$
Category:Historical Notes
1326

== Number ==
$1326$ (one thousand, three hundred and twenty-six) is:
:$2 \times 13 \times 51$
:The $5$th hexamorphic number after $1$, $45$, $66$, $1225$:
::$1326 = H_{26}$
:The $26$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $703$, $780$, $861$, $946$, $1035$, $1128$, $1225$:
::$1326 = \ds \sum_{k \mathop = 1}^{26} \paren {4 k - 3} = 26 \paren {2 \times 26 - 1}$
:The $51$st triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $990$, $1035$, $1081$, $1128$, $1176$, $1225$, $1275$:
::$1326 = \ds \sum_{k \mathop = 1}^{51} k = \dfrac {51 \times \paren {51 + 1} } 2$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:1326
132 is Sum of all 2-Digit Numbers formed from its Digits

== Theorem ==
$132$ is the smallest sum of all the $2$-digit (positive) integers formed from its own digits.
== Proof ==
It is necessary to postulate that such (positive) integers have $3$ digits or more, as the $2$ digit solution is trivial.
Let $n = \sqbrk {abc}$ be a $3$-digit number.
Let $\map s n$ denote the sum of all the $2$-digit (positive) integers formed from the digits of $n$.
Then:
So for $n = \map s n$, $n = \sqbrk {abc}$ needs to be divisible by both $22$ and $a + b + c$.
and the result is apparent.
== Sources ==
* : $132$
* : $132$
Category:132
133

== Number ==
$133$ (one hundred and thirty-three) is:
:$7 \times 19$
:One of the prime factors of one of the terms of one of the $2$ representations of $635 \, 318 \, 657$ as the sum of $2$ fourth powers:
::$635 \, 318 \, 657 = 133^4 + 134^4$
:The $7$th octagonal number, after $1$, $8$, $21$, $40$, $65$, $96$:
::$133 = 1 + 7 + 13 + 19 + 25 + 31 + 37 = 7 \paren {3 \times 7 - 2}$
:The $25$th happy number after $1$, $7$, $10$, $13$, $19$, $23$, $\ldots$, $91$, $94$, $97$, $100$, $103$, $109$, $129$, $130$:
::$133 \to 1^2 + 3^2 + 3^2 = 1 + 9 + 9 = 19 \to 1^2 + 9^2 = 1 + 81 = 82 \to 8^2 + 2^2 = 64 + 4 = 68 \to 6^2 + 8^2 = 36 + 64 = 100 \to 1^2 + 0^2 + 0^2 = 1$
:The $28$th lucky number:
::$1$, $3$, $7$, $9$, $13$, $15$, $21$, $25$, $31$, $33$, $37$, $43$, $49$, $51$, $63$, $67$, $73$, $75$, $79$, $87$, $93$, $99$, $105$, $111$, $115$, $127$, $129$, $133$, $\ldots$
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:133
133,496

== Number ==
$133 \, 496$ (one hundred and thirty-three thousand, four hundred and ninety-six) is:
:$2^3 \times 11 \times 37 \times 41$
:The $9$th subfactorial after $0$, $1$, $2$, $9$, $44$, $265$, $1854$, $14 \, 833$:
::$133 \, 496 = \, !9 = 9! \left({1 - \dfrac 1 {1!} + \dfrac 1 {2!} - \dfrac 1 {3!} + \dfrac 1 {4!} - \dfrac 1 {5!} + \dfrac 1 {6!} - \dfrac 1 {7!} + \dfrac 1 {8!} - \dfrac 1 {9!} }\right)$
== Also see ==
* 
Category:Specific Numbers
Category:133,496
1330

== Number ==
$1330$ (one thousand, three hundred and thirty) is:
:$2 \times 5 \times 7 \times 19$
:The $19$th tetrahedral number, after $1$, $4$, $10$, $20$, $35$, $56$, $84$, $120$, $165$, $220$, $286$, $364$, $455$, $560$, $680$, $816$, $969$, $1140$:
::$1330 = \ds \sum_{k \mathop = 1}^{19} \frac {k \paren {k + 1} } 2 = \dfrac {19 \paren {19 + 1} \paren {19 + 2} } 6$
:The $23$rd primitive semiperfect number after $6$, $20$, $28$, $88$, $104$, $\ldots$, $572$, $650$, $748$, $770$, $910$, $945$, $1184$, $1190$, $1312$:
::$1330 = 2 + 5 + 7 + 14 + 35 + 38 + 70 + 95 + 133 + 266 + 665$
== Also see ==
* 
* 
Category:Specific Numbers
Category:1330
1331

== Number ==
$1331$ (one thousand, three hundred and thirty-one) is:
:$11^3$
:The $3$rd power of $11$ after $(1)$, $11$, $121$:
::$1331 = 11^3$
:The $4$th palindromic cube after $1$, $8$, $343$:
::$1331 = 11^3$
:The $11$th cube number after $1$, $8$, $27$, $64$, $125$, $216$, $343$, $512$, $729$, $1000$:
::$1331 = 11 \times 11 \times 11$
:A cube number which is the sum of $3$ consecutive odd squares:
::$1331 = 11^3 = 19^2 + 21^2 + 23^2$
== Also see ==
* Cube of 11 is Palindromic
* Cube Number as Sum of Three Consecutive Odd Squares
* 
* 
* 
== Sources ==
* : $121$
* : $121$
* : $1331$
Category:Specific Numbers
Category:1331
1332

== Number ==
$1332$ (one thousand three hundred and thirty-two) is:
:$2^2 \times 3^2 \times 11$
:The $47$th Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$, $12$, $\ldots$, $1111$, $1112$, $1113$, $1115$, $1116$, $1131$, $1176$, $1184$, $1197$, $1212$, $1296$, $1311$:
::$1332 = 74 \times 18 = 74 \times \left({1 \times 3 \times 3 \times 2}\right)$
== Also see ==
* 
Category:Specific Numbers
Category:1332
1335

== Number ==
$1335$ (one thousand, three hundred and thirty-five) is:
:$3 \times 5 \times 89$
:The $30$th pentagonal number after $1$, $5$, $12$, $22$, $35$, $\ldots$, $477$, $532$, $590$, $651$, $715$, $782$, $852$, $925$, $1001$, $1080$, $1162$, $1247$:
::$1335 = \ds \sum_{k \mathop = 1}^{30} \paren {3 k - 2} = \dfrac {30 \paren {3 \times 30 - 1} } 2$
:The $59$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $925$, $950$, $1001$, $1027$, $1080$, $1107$, $1162$, $1190$, $1247$, $1276$:
::$1335 = \ds \sum_{k \mathop = 1}^{30} \paren {3 k - 2} = \dfrac {30 \paren {3 \times 30 - 1} } 2$
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Category:Specific Numbers
Category:1335
134

== Number ==
$134$ (one hundred and thirty-four) is:
:$2 \times 67$
:One of the prime factors of one of the terms of one of the $2$ representations of $635 \, 318 \, 657$ as the sum of $2$ fourth powers:
::$635 \, 318 \, 657 = 133^4 + 134^4$
:The $12$th noncototient after $10$, $26$, $34$, $50$, $52$, $58$, $86$, $100$, $116$, $122$, $130$:
::$\nexists m \in \Z_{>0}: m - \map \phi m = 134$
:where $\map \phi m$ denotes the Euler $\phi$ function
:The $18$th nontotient after $14$, $26$, $34$, $38$, $50$, $62$, $68$, $74$, $76$, $86$, $90$, $94$, $98$, $114$, $118$, $122$, $124$:
::$\nexists m \in \Z_{>0}: \map \phi m = 134$
:where $\map \phi m$ denotes the Euler $\phi$ function
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Category:Specific Numbers
Category:134
1344

== Number ==
$1344$ (one thousand three hundred and forty-four) is:
:$2^6 \times 3 \times 7$
:The $48$th Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$, $12$, $\ldots$, $1176$, $1184$, $1197$, $1212$, $1296$, $1311$, $1332$:
::$1344 = 28 \times 48 = 28 \times \left({1 \times 3 \times 4 \times 4}\right)$
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Category:Specific Numbers
Category:1344
135

== Number ==
$135$ (one hundred and thirty-five) is:
:$3^3 \times 5$
:The $6$th of the $17$ positive integers for which the value of the Euler $\phi$ function is $72$:
::$73$, $91$, $95$, $111$, $117$, $135$, $146$, $148$, $152$, $182$, $190$, $216$, $222$, $228$, $234$, $252$, $270$
:The $12$th number after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $89$ which is the sum of the increasing powers of its digits taken in order:
::$1^1 + 3^2 + 5^3 = 135$
:The number of integer partitions for $14$:
::$\map p {14} = 135$
:The $20$th Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$, $12$, $15$, $24$, $36$, $111$, $112$, $115$, $128$, $132$:
::$135 = 9 \times 15 = 9 \times \left({1 \times 3 \times 5}\right)$
:The $26$th positive integer $n$ such that no factorial of an integer can end with $n$ zeroes.
:The $29$th lucky number:
::$1$, $3$, $7$, $9$, $13$, $15$, $21$, $\ldots$, $105$, $111$, $115$, $127$, $129$, $133$, $135$, $\ldots$
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* : $135$
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Category:Specific Numbers
Category:135
1351

== Number ==
$1351$ (one thousand, three hundred and fifty-one) is:
:$7 \times 193$
:The $15$th non-square positive integer which cannot be expressed as the sum of a square and a prime:
::$10$, $34$, $58$, $85$, $91$, $130$, $214$, $226$, $370$, $526$, $706$, $730$, $771$, $1255$, $1351$, $\ldots$
:The $32$nd positive integer which cannot be expressed as the sum of a square and a prime:
::$1$, $10$, $25$, $34$, $58$, $64$, $85$, $\ldots$, $706$, $730$, $771$, $784$, $841$, $1024$, $1089$, $1225$, $1255$, $1351$, $\ldots$
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Category:Specific Numbers
Category:1351
1354

== Number ==
$1354$ (one thousand, three hundred and fifty-four) is:
:$2 \times 677$
:The $27$th even integer after $2$, $4$, $94$, $96$, $\ldots$, $1148$, $1264$, $1266$, $1268$ that cannot be expressed as the sum of $2$ prime numbers which are each one of a pair of twin primes
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Category:Specific Numbers
Category:1354
1356

== Number ==
$1356$ (one thousand, three hundred and fifty-six) is:
:$2^2 \times 3 \times 113$
:The $28$th even integer after $2$, $4$, $94$, $96$, $\ldots$, $1148$, $1264$, $1266$, $1268$, $1354$ that cannot be expressed as the sum of $2$ prime numbers which are each one of a pair of twin primes
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Category:Specific Numbers
Category:1356
1358

== Number ==
$1358$ (one thousand, three hundred and fifty-eight) is:
:$2 \times 7 \times 97$
:The $29$th even integer after $2$, $4$, $94$, $96$, $\ldots$, $1148$, $1264$, $1266$, $1268$, $1354$, $1356$ that cannot be expressed as the sum of $2$ prime numbers which are each one of a pair of twin primes
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Category:Specific Numbers
Category:1358