nyuuzyou/wikis huggingface dataset:
475

== Number ==
$475$ (four hundred and seventy-five) is:
:$5^2 \times 19$
:The $21$st integer after $7$, $13$, $19$, $35$, $38$, $\ldots$, $223$, $253$, $285$, $305$, $350$, $380$, $410$ the decimal representation of whose square can be split into two parts which are each themselves square:
::$475^2 = 225 \, 625$; $225 = 15^2$, $625 = 25^2$
== Also see ==
* 
Category:Specific Numbers
Category:475
4753

== Number ==
$4753$ (four thousand, seven hundred and fifty-three) is:
:$7^2 \times 97$
:The $49$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $3160$, $3321$, $3486$, $3655$, $3828$, $4005$, $4186$, $4371$, $4560$:
::$4753 = \ds \sum_{k \mathop = 1}^{49} \paren {4 k - 3} = 49 \paren {2 \times 49 - 1}$
:The $97$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $4095$, $4186$, $4278$, $4371$, $4465$, $4560$, $4656$:
::$4753 = \ds \sum_{k \mathop = 1}^{97} k = \dfrac {97 \times \paren {97 + 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:4753
4761

== Number ==
$4761$ (four thousand, seven hundred and sixty-one) is:
:$3^2 \times 23^2$
:The $69$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $3249$, $3364$, $3481$, $3600$, $3721$, $3844$, $3969$, $4096$, $4225$, $4356$, $4489$, $4624$:
::$4761 = 69 \times 69$
== Also see ==
* 
Category:Specific Numbers
Category:4761
477

== Number ==
$477$ (four hundred and seventy-seven) is:
:$3^2 \times 53$
:The $18$th pentagonal number after $1$, $5$, $12$, $22$, $35$, $51$, $70$, $92$, $117$, $145$, $176$, $210$, $247$, $287$, $330$, $330$, $376$, $425$:
::$477 = \ds \sum_{k \mathop = 1}^{18} \paren {3 k - 2} = \dfrac {18 \paren {3 \times 18 - 1} } 2$
:The $35$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $222$, $247$, $260$, $287$, $301$, $330$, $345$, $376$, $392$, $425$, $442$:
::$477 = \ds \sum_{k \mathop = 1}^{18} \paren {3 k - 2} = \dfrac {18 \paren {3 \times 18 - 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:477
4774

== Number ==
$4774$ (four thousand, seven hundred and seventy-four) is:
:$2 \times 7 \times 11 \times 31$
:The $44$th heptagonal number after $1$, $7$, $18$, $34$, $55$, $81$, $112$, $\ldots$, $3010$, $3186$, $3367$, $3744$, $3940$, $4141$, $4347$, $4558$:
::$4774 = \ds \sum_{k \mathop = 1}^{44} \paren {5 k - 4} = \dfrac {44 \paren {5 \times 44 - 3} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:4774
4787

== Number ==
$4787$ (four thousand, seven hundred and eighty-seven) is:
:The $643$rd prime number
:The $13$th prime $p$ such that $p \# + 1$, where $p \#$ denotes primorial (product of all primes up to $p$) of $p$, is prime:
::$2$, $3$, $5$, $7$, $11$, $31$, $379$, $1019$, $1021$, $2657$, $3229$, $4547$, $4787$
== Also see ==
* 
* 
Category:Specific Numbers
Category:4787
479

== Number ==
$479$ (four hundred and seventy-nine) is:
:The $92$nd prime number
:The smaller of the $6$th pair of primes whose prime gap is $8$:
::$487 - 479 = 8$
:The $6$th positive integer after $79$, $159$, $239$, $319$, $399$ which cannot be expressed as the sum of fewer than $19$ fourth powers:
::$479 = 13 \times 1^4 + 3 \times 2^4 + 2 \times 3^4 + 4^4$
:or:
::$479 = 10 \times 1^4 + 4 \times 2^4 + 5 \times 3^4$
:The $16$th of $29$ primes of the form $2 x^2 + 29$:
::$2 \times 15^2 + 29 = 479$ ()
:The $17$th safe prime after $5$, $7$, $11$, $23$, $47$, $59$, $83$, $107$, $167$, $179$, $227$, $263$, $347$, $359$, $383$, $467$:
::$479 = 2 \times 239 + 1$
== Also see ==
* 
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Category:Specific Numbers
Category:479
479 is not Expressible as Sum of Fewer than 19 Fourth Powers

== Theorem ==
$479$ cannot be expressed as the sum of fewer than $19$ fourth powers:
:$479 = 13 \times 1^4 + 3 \times 2^4 + 2 \times 3^4 + 4^4$
or:
:$479 = 10 \times 1^4 + 4 \times 2^4 + 5 \times 3^4$
== Proof ==
First note that $5^4 = 625 > 479$.
Then note that $2 \times 4^4 = 512 > 479$.
Hence any expression of $479$ as fourth powers uses no $n^4$ for $n \ge 5$, and uses not more than $1$ instance of $4^4$.
For the remainder, using $2^4$ uses fewer fourth powers than $16$ instances of $1^4$ does.
Now we have:
and it can be seen that the first and the fourth uses the least number of fourth powers, at $19$.
== Also see ==
* Hilbert-Waring Theorem
* Numbers not Expressible as Sum of Fewer than 19 Fourth Powers
Category:Fourth Powers
Category:Hilbert-Waring Theorem
Category:479
48

== Number ==
$48$ (forty-eight) is:
:$2^4 \times 3$
:The $1$st composite number the product of whose proper divisors form its $4$th power:
::$1 \times 2 \times 3 \times 4 \times 6 \times 8 \times 12 \times 16 \times 24 = 48^4$
:With $75$, an element of the $1$st quasiamicable pair:
::$\map {\sigma_1} {48} = \map {\sigma_1} {75} = 124 = 48 + 75 + 1$
:The $2$nd of the largest known pair of Ulam numbers which differ by $1$:
::$47 = 11 + 36, \ 48 = 1 + 47$
:The smallest positive integer which can be expressed as the sum of $2$ odd primes in $5$ ways:
::$48 = 43 + 5 = 41 + 7 = 37 + 11 = 31 + 17 = 29 + 19$
:The $8$th highly composite number after $1$, $2$, $4$, $6$, $12$, $24$, $36$:
::$\map {\sigma_0} {48} = 10$
:The $8$th superabundant number after $1$, $2$, $4$, $6$, $12$, $24$, $36$:
::$\dfrac {\map {\sigma_1} {48} } {48} = \dfrac {124} {48} = 2 \cdotp 58 \dot 3$
:The $9$th positive integer $n$ after $5$, $11$, $17$, $23$, $29$, $30$, $36$, $42$ such that no factorial of an integer can end with $n$ zeroes.
:The $11$th semiperfect number after $6$, $12$, $18$, $20$, $24$, $28$, $30$, $36$, $40$, $42$:
::$48 = 8 + 16 + 24$
:The $16$th Ulam number after $1$, $2$, $3$, $4$, $6$, $8$, $11$, $13$, $16$, $18$, $26$, $28$, $36$, $38$, $47$:
::$48 = 1 + 47$
:The $16$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $12$, $16$, $18$, $20$, $24$, $30$, $36$, $42$:
::$\map {\sigma_1} {48} = 124$
:The $18$th of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:
::$3$, $4$, $9$, $10$, $12$, $16$, $17$, $22$, $23$, $25$, $27$, $29$, $30$, $36$, $38$, $40$, $43$, $48$, $\ldots$
:The $18$th of $21$ integers which can be represented as the sum of two primes in the maximum number of ways
::$1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $10$, $12$, $14$, $16$, $18$, $24$, $30$, $36$, $42$, $48$, $\ldots$
:The $22$nd after $1$, $2$, $4$, $5$, $6$, $8$, $9$, $12$, $13$, $15$, $16$, $17$, $20$, $24$, $25$, $27$, $28$, $32$, $35$, $26$, $39$ of the $24$ positive integers which cannot be expressed as the sum of distinct non-pythagorean primes
:The $22$nd positive integer which is not the sum of $1$ or more distinct squares:
::$2$, $3$, $6$, $7$, $8$, $11$, $12$, $15$, $18$, $19$, $22$, $23$, $24$, $27$, $28$, $31$, $32$, $33$, $43$, $44$, $47$, $48$, $\ldots$
:The $32$nd (strictly) positive integer after $1$, $2$, $3$, $\ldots$, $27$, $30$, $31$, $32$, $35$, $36$, $37$, $38$, $42$, $43$, $44$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$
== Also see ==
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== Sources ==
* : $48$
* : $48$
Category:Specific Numbers
Category:48
48,976

== Number ==
$48 \, 976$ (forty-eight thousand, nine hundred and seventy-six) is:
:$2^4 \times 3061$
:The $23$rd 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:48,976
48,988,659,276,962,496

== Number ==
$48 \, 988 \, 659 \, 276 \, 962 \, 496$ is:
:$2^6 \times 3^3 \times 7^4 \times 13 \times 19 \times 43 \times 73 \times 97 \times 157$
:The $5$th Hardy-Ramanujan number after $2$, $1729$, $87 \, 539 \, 319$, $6 \, 963 \, 472 \, 309 \, 248$: the smallest positive integer which can be expressed as the sum of $2$ cubes in $5$ different ways:
== Also see ==
* 
== Sources ==
* : $48,988,659,276,962,496$
Category:Specific Numbers
Category:48,988,659,276,962,496
480

== Number ==
$480$ (four hundred and eighty) is:
:$2^5 \times 3 \times 5$
:The $9$th positive integer after $128$, $192$, $256$, $288$, $320$, $384$, $432$, $448$ with $7$ or more prime factors:
::$480 = 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 5$
:The $22$nd positive integer after $64$, $96$, $128$, $144$, $\ldots$, $320$, $324$, $336$, $352$, $360$, $384$, $400$, $416$, $432$, $448$ with $6$ or more prime factors:
::$480 = 2 \times 2 \times 2 \times 2 \times 2 \times 3 \paren {\times \, 5}$
:The $36$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $\ldots$, $120$, $144$, $168$, $180$, $210$, $216$, $240$, $288$, $300$, $336$, $360$, $420$:
::$\map {\sigma_1} {480} = 1512$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:480
481

== Number ==
$481$ (four hundred and eighty-one) is:
:$13 \times 37$
:The smallest positive integer whose square is the sum of $3$ fourth powers:
::$481^2 = 12^4 + 15^4 + 20^4$
:The $13$th octagonal number, after $1$, $8$, $21$, $40$, $65$, $96$, $133$, $176$, $225$, $280$, $341$, $408$:
::$\ds 481 = \sum_{k \mathop = 1}^{13} \paren {6 k - 5} = 13 \paren {3 \times 13 - 2}$
== Also see ==
* Smallest Square which is Sum of 3 Fourth Powers
* 
== Sources ==
* : $481$
Category:Specific Numbers
Category:481
482

== Number ==
$482$ (four hundred and eighty-two) is:
:$2 \times 241$
:The $47$th noncototient after $10$, $26$, $34$, $50$, $\ldots$, $344$, $346$, $362$, $366$, $372$, $386$, $394$, $404$, $412$, $436$, $466$, $470$, $474$:
::$\nexists m \in \Z_{>0}: m - \map \phi m = 482$
:where $\map \phi m$ denotes the Euler $\phi$ function
== Also see ==
* 
Category:Specific Numbers
Category:482
483

== Number ==
$483$ (four hundred and eighty-three) is:
:$3 \times 7 \times 23$
:The $20$th Smith number after $4$, $22$, $27$, $58$, $\ldots$, $346$, $355$, $378$, $382$, $391$, $438$, $454$:
::$4 + 8 + 3 = 3 + 7 + 2 + 3 = 15$
== Also see ==
* 
Category:Specific Numbers
Category:483
484

== Number ==
$484$ (four hundred and eighty-four) is:
:$2^2 \times 11^2$
:The $1$st of the only $3$ square numbers which start and end with $n$ $4$s.
:The $10$th integer after $0$, $1$, $2$, $4$, $8$, $121$, $151$, $212$, $242$ which is palindromic in both decimal and ternary:
::$484_{10} = 122 \, 221_3$
:The $14$th square after $1$, $4$, $9$, $16$, $25$, $36$, $49$, $64$, $81$, $121$, $144$, $225$, $441$ which has no more than $2$ distinct digits and does not end in $0$:
::$484 = 22^2$
:The $22$nd square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $225$, $256$, $289$, $324$, $361$, $400$, $441$:
::$484 = 22 \times 22$
:The $37$th powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $\ldots$, $289$, $324$, $343$, $361$, $392$, $400$, $432$, $441$:
::$484 = 2^2 \times 11^2$
:The palindromic square of the palindromic number $22$:
== Also see ==
* Square of Small-Digit Palindromic Number is Palindromic
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== Sources ==
* : $484$
* : $484$
Category:Specific Numbers
Category:484
485

== Number ==
$485$ (four hundred and eighty-five) is:
:$5 \times 97$
:The $20$th positive integer $n$ after $0$, $1$, $5$, $25$, $29$, $41$, $49$, $61$, $65$, $85$, $89$, $101$, $125$, $145$, $149$, $245$, $265$, $365$, $385$ such that the Fibonacci number $F_n$ ends in $n$
== Also see ==
* 
Category:Specific Numbers
Category:485
4851

== Number ==
$4851$ (four thousand, eight hundred and fifty-one) is:
:$3^2 \times 7^2 \times 11$
:The $21$st pentagonal pyramidal number after $1$, $6$, $12$, $40$, $75$, $126$, $196$, $288$, $405$, $550$, $726$, $936$, $1183$, $1470$, $1800$, $2176$, $2601$, $3078$, $3610$, $4200$:
::$4851 = \ds \sum_{k \mathop = 1}^{21} \dfrac {k \paren {3 k - 1} } 2 = \dfrac {21^2 \paren {21 + 1} } 2$
:The $98$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $4095$, $4186$, $4278$, $4371$, $4465$, $4560$, $4656$, $4753$:
::$4851 = \ds \sum_{k \mathop = 1}^{98} k = \dfrac {98 \times \paren {98 + 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:4851
486

== Number ==
$486$ (four hundred and eighty-six) is:
:$2 \times 3^5$
:The $23$rd positive integer after $64$, $96$, $128$, $144$, $\ldots$, $384$, $400$, $416$, $432$, $448$, $480$ with $6$ or more prime factors:
::$486 = 2 \times 3 \times 3 \times 3 \times 3 \times 3$
== Also see ==
* 
Category:Specific Numbers
Category:486
4862

== Number ==
$4862$ (four thousand, eight hundred and sixty-two) is:
:$2 \times 11 \times 13 \times 17$
:The $9$th Catalan number after $(1)$, $1$, $2$, $5$, $14$, $42$, $132$, $429$, $1430$:
::$4862 = \dfrac 1 {9 + 1} \dbinom {2 \times 9} 9 = \dfrac 1 {10} \times 48 \, 620$
== Also see ==
* 
Category:Specific Numbers
Category:4862
487

== Number ==
$487$ (four hundred and eighty-seven) is:
:The $93$rd prime number
:The $2$nd prime number $p$ after $3$ which satisfies the equation $p^2 \divides \paren {10^p - 10}$:
::$487^2 \divides \paren {10^{487} - 10}$
:The $6$th positive integer $n$ after $157$, $262$, $367$, $412$, $472$ for which $\map \phi {2 n + 1} < \map \phi {2 n}$:
::$\map \phi {2 \times 487 + 1} = 480, \map \phi {2 \times 487} = 486$
:The larger of the $6$th pair of primes whose prime gap is $8$:
::$487 - 479 = 8$
:The $22$nd integer after $7$, $13$, $19$, $35$, $38$, $\ldots$, $223$, $253$, $285$, $305$, $350$, $380$, $410$, $475$ the decimal representation of whose square can be split into two parts which are each themselves square:
::$487^2 = 237 \, 169$; $23 \, 716 = 154^2$, $9 = 3^2$
:The $33$rd long period prime after $7$, $17$, $19$, $23$, $29$, $\ldots$, $269$, $313$, $337$, $367$, $379$, $383$, $389$, $419$, $433$, $461$
== Also see ==
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== Sources ==
* : $487$
Category:Specific Numbers
Category:487
4879

== Number ==
$4879$ (four thousand, eight hundred and seventy-nine) is:
:$7 \times 17 \times 41$
:The $11$th Kaprekar number after $1$, $9$, $45$, $55$, $99$, $297$, $703$, $999$, $2223$, $2728$:
::$4879^2 = 23 \, 804 \, 641 \to 238 + 04641 = 4879$
== Also see ==
* 
Category:Specific Numbers
Category:4879
49

== Number ==
$49$ (forty-nine) is:
:$7^2$
:The $1$st square whose decimal representation can be split into two parts which are each themselves square:
::$49 = 7^2$; $4 = 2^2$, $9 = 3^2$
:The $1$st term of the $1$st $5$-tuple of consecutive integers have the property that they are not values of the divisor sum function $\map {\sigma_1} n$ for any $n$:
::$\tuple {49, 50, 51, 52, 53}$
:The $2$nd power of $7$ after $(1)$, $7$:
::$49 = 7^2$
:The $4$th square lucky number:
::$1$, $9$, $25$, $49$, $\ldots$
:The $7$th square number after $1$, $4$, $9$, $16$, $25$, $36$:
::$49 = 7 \times 7$
:The $7$th square after $1$, $4$, $9$, $16$, $25$, $36$ which has no more than $2$ distinct digits and does not end in $0$:
::$49 = 7^2$
:The $7$th positive integer $n$ after $0$, $1$, $5$, $25$, $29$, $41$ such that the Fibonacci number $F_n$ ends in $n$
:The $8$th trimorphic number after $1$, $4$, $5$, $6$, $9$, $24$, $25$:
::$49^3 = 117 \, 6 \mathbf {49}$
:The $10$th powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $27$, $32$, $36$
:The $11$th happy number after $1$, $7$, $10$, $13$, $19$, $23$, $28$, $31$, $32$, $44$:
::$49 \to 4^2 + 9^2 = 16 + 81 = 97 \to 9^2 + 7^2 = 81 + 49 = 130 \to 1^2 + 3^2 + 0^2 = 1 + 9 + 0 = 10 \to 1^2 + 0^2 = 1$
:The $13$th lucky number:
::$1$, $3$, $7$, $9$, $13$, $15$, $21$, $25$, $31$, $33$, $37$, $43$, $49$, $\ldots$
:The $17$th semiprime after $4$, $6$, $9$, $10$, $14$, $15$, $21$, $22$, $25$, $26$, $33$, $34$, $35$, $38$, $39$, $46$:
::$49 = 7 \times 7$
:The $24$th odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
::$1$, $3$, $5$, $7$, $\ldots$, $35$, $37$, $41$, $43$, $45$, $47$, $49$, $\ldots$
:The $29$th integer $n$ such that $2^n$ contains no zero in its decimal representation:
::$2^{49} = 562 \, 949 \, 953 \, 421 \, 312$
:The $30$th positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $33$, $37$, $38$, $42$, $43$, $44$, $45$, $46$ which cannot be expressed as the sum of distinct pentagonal numbers.
:The $33$rd (strictly) positive integer after $1$, $2$, $3$, $\ldots$, $37$, $38$, $42$, $43$, $44$, $48$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$
== Also see ==
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== Sources ==
* : $49$
* : $49$
Category:Specific Numbers
Category:49
49,081

== Number ==
$49 \, 081$ (forty-nine thousand and eighty-one) is:
:The $5045$th prime number
:The index of the $6$th (probable) repunit prime after $R_2$, $R_{19}$, $R_{23}$, $R_{317}$, $R_{1031}$
== Also see ==
* 
* 
Category:Specific Numbers
Category:49,081
49,463,435,743,205,655

== Number ==
$49 \, 463 \, 435 \, 743 \, 205 \, 655$ is:
:$3 \times 5 \times 83 \times 89 \times 239 \times 1 \, 867 \, 785 \, 589$
:The $2$nd term of a general Fibonacci sequence whose first $2$ terms are coprime such that all its terms are composite:
::$62 \, 638 \, 280 \, 004 \, 239 \, 857$, $49 \, 463 \, 435 \, 743 \, 205 \, 655$, $112 \, 101 \, 715 \, 747 \, 445 \, 512$, $\ldots$
== Also see ==
* 
== Sources ==
* : $62,638,280,004,239,857$
Category:Specific Numbers
Category:49,463,435,743,205,655
490

== Number ==
$490$ (four hundred and ninety) is:
:$2 \times 5 \times 7^2$
:The $11$th primitive semiperfect number after $6$, $20$, $28$, $88$, $104$, $272$, $304$, $350$, $368$, $464$:
::$490 = 2 + 5 + 7 + 14 + 49 + 70 + 98 + 245$
:The number of integer partitions for $19$:
::$\map p {19} = 490$
:The smallest positive integer which can be expressed as the sum of $2$ distinct lucky numbers in $23$ different ways
:The $48$th noncototient after $10$, $26$, $34$, $50$, $\ldots$, $386$, $394$, $404$, $412$, $436$, $466$, $470$, $474$, $482$:
::$\nexists m \in \Z_{>0}: m - \map \phi m = 490$
:where $\map \phi m$ denotes the Euler $\phi$ function
== Also see ==
* 
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Category:Specific Numbers
Category:490
490,689

== Number ==
$490 \, 689$ (four hundred and ninety thousand, six hundred and eighty-nine) is:
:$3^2 \times 54521$
:The sum of $3$ cubes in $2$ different ways:
::$490 \, 689 = 4^3 + 60^3 + 65^3 = 8^3 + 25^3 \times 78^3$
:while at the same time $4 \times 60 \times 65 = 8 \times 25 \times 78$
== Also see ==
* Property of 490,689
== Sources ==
* : $490,689$
Category:Specific Numbers
Category:490,689
4900

== Number ==
$4900$ (four thousand, nine hundred) is:
:$2^2 \times 5^2 \times 7^2$
:The only square pyramidal number which is also square:
::$4900 = 70^2 = \ds \sum_{k \mathop = 1}^{24} k^2 = \dfrac {24 \paren {24 + 1} \paren {2 \times 24 + 1} } 6$
:The $9$th square after $49$, $169$, $361$, $1225$, $1444$, $1681$, $3249$, $4225$ whose decimal representation can be split into two parts which are each themselves square:
::$4900 = 70^2$; $4 = 2^2$, $900 = 30^2$
:The $24$th square pyramidal number after $1$, $5$, $14$, $30$, $55$, $\ldots$, $1015$, $1240$, $1496$, $1785$, $2109$, $2470$, $2870$, $3311$, $3795$, $4324$:
::$4900 = \ds \sum_{k \mathop = 1}^{24} k^2 = \dfrac {24 \paren {24 + 1} \paren {2 \times 24 + 1} } 6$
:The $27$th square number after $1$, $4$, $36$, $121$, $144$, $256$, $\ldots$, $1936$, $2304$, $2704$, $2916$, $3136$, $3600$, $3844$, $4096$, $4356$, $4624$ to be the divisor sum value of some (strictly) positive integer:
::$4900 = \map {\sigma_1} {4537}$
:The $70$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $4096$, $4225$, $4356$, $4489$, $4624$, $4761$:
::$4900 = 70 \times 70$
== Also see ==
* Square Pyramidal Number also Square
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== Sources ==
* : $4900$
* : $4900$
Category:Specific Numbers
Category:4900
491

== Number ==
$491$ (four hundred and ninety-one) is:
:The $94$th prime number
:The smaller of the $7$th pair of primes whose prime gap is $8$:
::$499 - 491 = 8$
:The $25$th Sophie Germain prime after $2$, $3$, $5$, $11$, $23$, $29$, $41$, $53$, $83$, $89$, $113$, $131$, $173$, $179$, $191$, $233$, $239$, $251$, $281$, $293$, $359$, $419$, $431$, $443$:
::$2 \times 491 + 1 = 983$, which is prime.
:The $34$th long period prime after $7$, $17$, $19$, $23$, $29$, $\ldots$, $269$, $313$, $337$, $367$, $379$, $383$, $389$, $419$, $433$, $461$, $487$
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:491
4913

== Number ==
$4913$ (four thousand, nine hundred and thirteen) is:
:$17^3$
:The $4$th cube after $0$, $1$, $512$ whose digits add up to its root:
::$4913 = 17^3$, while $4 + 9 + 1 + 3 = 17$
:The $17$th cube number after $1$, $8$, $27$, $64$, $125$, $216$, $343$, $512$, $729$, $1000$, $1331$, $1728$, $2197$, $2744$, $3375$, $4096$:
::$4913 = 17 \times 17 \times 17$
== Also see ==
* 
* 
== Sources ==
* : $4913$
* : $4913$
Category:Specific Numbers
Category:4913
492

== Number ==
$492$ (four hundred and ninety-two) is:
:$2^2 \times 3 \times 41$
:The sum of $3$ cubes, either positive or negative, in $3$ known ways:
:Its cube can be expressed as the sum of $3$ positive cubes in $13$ different ways: 
== Also see ==
* 492 is Sum of 3 Cubes in 3 Ways
* 492 Cubed is Sum of 3 Positive Cubes in 13 Ways
== Sources ==
* : $492$
Category:Specific Numbers
Category:492
4921

== Number ==
$4921$ (four thousand, nine hundred and twenty-one) is:
:$7 \times 19 \times 37$
:The $41$st centered hexagonal number after $1$, $7$, $19$, $37$, $61$, $91$, $127$, $\ldots$, $2791$, $2977$, $3169$, $3367$, $3571$, $3781$, $3997$, $4219$, $4447$, $4681$:
::$4921 = \ds 1 + \sum_{k \mathop = 1}^{41 - 1} 6 k = 41^3 - 40^3$
== Also see ==
* 
Category:Specific Numbers
Category:4921
492 Cubed is Sum of 3 Positive Cubes in 13 Ways

== Theorem ==
The cube of $492$ can be expressed as the sum of $3$ positive cubes in $13$ different ways:
== Proof ==
Brute force.
== Also see ==
* 492 is Sum of 3 Cubes in 3 Ways
== Historical Note ==
== Sources ==
* 
* 
* : $6$: Number Recreations: Narcissistic Numbers: Miscellaneous
* : $492$
Category:492
Category:Sums of Cubes
492 Cubed is Sum of 3 Positive Cubes in 13 Ways/Historical Note

== Historical Note on 492 Cubed is Sum of 3 Positive Cubes in 13 Ways ==
The question was asked in of January $1957$ for a (positive) integer less than $1000$ whose cube could be expressed as the sum of $3$ cubes in $5$ distinct ways.
A solution submitted by returned some $47$ such integers which have between $5$ and $9$ such expressions, but surprisingly $492$ was not among them.
$870$ had most such representations, that is, $9$.
Subsequent investigation unearthed several more.
By the time reported in his of $1966$, there were $2$ integers each with $10$ such representations (that is, $492$ and $870$), and one had $11$ (that is, $792$).
The representations for $492$ and $792$ were found by , while the representations for $870$ were, as for the $1957$ investigation, attributed to .
A more recent investigation, using a computer program designed to test exhaustively all integers less than $1000$ has revealed that each of $492$, $792$ and $870$ has exactly $13$ such representations.
== Sources ==
* 
* 
* : $6$: Number Recreations: Narcissistic Numbers: Miscellaneous
Category:Historical Notes
492 Cubed is Sum of 3 Positive Cubes in 13 Ways/Mistake

== Source Work ==
* 
:$6$: Number Recreations
::Narcissistic Numbers:
:::Miscellaneous
== Mistake ==
:... an integer less than $1,000$ whose cube could be represented in $5$ distinct ways as the sum of the cubes of $3$ positive integers...
:In the table below, three solutions to the problem are shown that have gone far beyond ...
::$\begin {array} {ccc}
& n = 492 & \\
a & b & c \\
24 & 204 & 480 \\
48 & 85 & 491 \\
72 & 384 & 396 \\
113 & 264 & 463 \\
144 & 360 & 414 \\
176 & 204 & 472 \\
207 & 297 & 438 \\
226 & 332 & 414 \\
246 & 328 & 410 \\
281 & 322 & 399 \\ 
\end {array}$
== Correction ==
The $5$th triple is wrong.
The first number is $114$, not $144$.
== Sources ==
* : $6$: Number Recreations: Narcissistic Numbers: Miscellaneous
Category:Mistakes/Mathematics on Vacation
492 is Sum of 3 Cubes in 3 Ways

== Theorem ==
$492$ can be expressed as the sum of $3$ cubes, either positive or negative in $3$ known ways.
== Proof ==
Brute force.
== Also see ==
* 492 Cubed is Sum of 3 Positive Cubes in 13 Ways
== Historical Note ==
== Sources ==
* : $492$
Category:492
Category:Sums of Cubes