nyuuzyou/wikis huggingface dataset:
526

== Number ==
$526$ (five hundred and twenty-six) is:
:$2 \times 263$
:The $10$th non-square positive integer which cannot be expressed as the sum of a square and a prime:
::$10$, $34$, $58$, $85$, $91$, $130$, $214$, $226$, $370$, $526$, $\ldots$
:The $19$th positive integer which cannot be expressed as the sum of a square and a prime:
::$1$, $10$, $25$, $34$, $58$, $64$, $85$, $91$, $121$, $130$, $169$, $196$, $214$, $226$, $289$, $324$, $370$, $400$, $526$, $\ldots$
:The $22$nd Smith number after $4$, $22$, $27$, $58$, $\ldots$, $346$, $355$, $378$, $382$, $391$, $438$, $454$, $483$, $517$:
::$5 + 2 + 6 = 2 + 2 + 6 + 3 = 13$
:The smallest positive integer which can be expressed as the sum of $2$ distinct lucky numbers in $27$ different ways
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:526
527

== Number ==
$527$ (five hundred and twenty-seven) is:
:$17 \times 31$
:The smallest number $n$ that needs $6$ numbers less than $n$ such that the product of their factorials is square
:The $29$th number whose divisor sum is square:
:
== Also see ==
* Smallest n needing 6 Numbers less than n so that Product of Factorials is Square
* 
== Sources ==
* : $527$
Category:Specific Numbers
Category:527
528

== Number ==
$528$ (five hundred and twenty-eight) is:
:$2^4 \times 3 \times 11$
:The $26$th positive integer after $64$, $96$, $128$, $144$, $\ldots$, $384$, $400$, $416$, $432$, $448$, $480$, $504$, $512$ with $6$ or more prime factors:
::$528 = 2 \times 2 \times 2 \times 2 \times 3 \times 11$
:The $32$nd triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $325$, $351$, $378$, $406$, $435$, $465$, $496$:
::$528 = \ds \sum_{k \mathop = 1}^{32} k = \dfrac {32 \times \paren {32 + 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:528
528,529

== Number ==
$528 \, 529$ (five hundred and twenty-eight thousand, five hundred and twenty-nine) is:
:$727^2$
:The $727$th square number:
::$528 \, 529 = 727 \times 727$
:The $4$th square number after $8281$, $183 \, 184$, $328 \, 329$ whose digits form $2$ consecutive integers:
::$528 \, 529 = 727^2$
:The $3$rd square number after $183 \, 184$, $328 \, 329$ whose digits form $2$ consecutive increasing integers:
::$528 \, 529 = 727^2$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:528,529
5282

== Number ==
$5282$ (five thousand, two hundred and eighty-two) is:
:$2 \times 19 \times 139$
:The number of possible arrangements of $8$ rooks on a standard chessboard so that no rook is attacking any other rook
== Also see ==
* 8 Mutually Non-Attacking Rooks on Chessboard
== Sources ==
* : $5282$
Category:Specific Numbers
Category:5282
529

== Number ==
$529$ (five hundred and twenty-nine) is:
:$23^2$
:The $20$th positive integer which cannot be expressed as the sum of a square and a prime:
::$1$, $10$, $25$, $34$, $58$, $64$, $85$, $91$, $121$, $130$, $169$, $196$, $214$, $226$, $289$, $324$, $370$, $400$, $526$, $529$, $\ldots$
:The $23$rd square number after $1$, $4$, $8$, $9$, $16$, $25$, $\ldots$, $289$, $324$, $343$, $361$, $392$, $400$, $432$, $441$, $484$:
::$529 = 23 \times 23$
:The $40$th powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $\ldots$, $289$, $324$, $343$, $361$, $392$, $400$, $432$, $441$, $484$, $500$, $512$:
::$529 = 23^2$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:529
5292

== Number ==
$5292$ (five thousand, two hundred and ninety-two) is:
:$2^2 \times 3^3 \times 7^2$
:The $14$th Kaprekar number after $1$, $9$, $45$, $55$, $99$, $297$, $703$, $999$, $2223$, $2728$, $4879$, $4950$, $5050$:
::$5292^2 = 28 \, 005 \, 264 \to 28 + 005264 = 5292$
== Also see ==
* 
Category:Specific Numbers
Category:5292
53

== Number ==
$53$ (fifty-three) is:
:The $16$th prime number, after $2$, $3$, $5$, $7$, $11$, $13$, $17$, $19$, $23$, $29$, $31$, $37$, $41$, $43$, $47$
:The $1$st prime number which cannot be expressed as either the sum of or the difference between a power of $2$ and a power of $3$.
:The $1$st prime number $p$ the period of whose reciprocal is $\dfrac {p - 1} 4$:
::$\dfrac 1 {53} = 0 \cdotp \dot 01886 \, 79245 \, 28 \dot 3$
:The $2$nd balanced prime after $5$:
::$53 = \dfrac {47 + 59} 2$
:The $5$th 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 $7$th two-sided prime after $2$, $3$, $5$, $7$, $23$, $37$:
::$53$, $5$, $3$ are prime
:The $8$th Sophie Germain prime after $2$, $3$, $5$, $11$, $23$, $29$, $41$:
::$2 \times 53 + 1 = 107$, which is prime.
:The $8$th prime $p$ after $11$, $23$, $29$, $37$, $41$, $43$, $47$ such that the Mersenne number $2^p - 1$ is composite
:The $9$th right-truncatable prime after $2$, $3$, $5$, $7$, $23$, $29$, $31$, $37$
:The $11$th left-truncatable prime after $2$, $3$, $5$, $7$, $13$, $17$, $23$, $37$, $43$, $47$
:The $17$th Ulam number after $1$, $2$, $3$, $4$, $6$, $8$, $11$, $13$, $16$, $18$, $26$, $28$, $36$, $38$, $47$, $48$:
::$53 = 6 + 47$
:The $20$th of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:
::$3$, $4$, $9$, $10$, $12$, $16$, $17$, $22$, $23$, $25$, $27$, $29$, $30$, $36$, $38$, $40$, $43$, $48$, $51$, $53$, $\ldots$
:The $25$th odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
::$1$, $3$, $5$, $7$, $\ldots$, $35$, $37$, $41$, $43$, $45$, $47$, $49$, $53$, $\ldots$
:The probability that out of $53$ people, no $2$ of them share the same birthday is approximately $\dfrac 1 {53}$.
== Also see ==
* Period of Reciprocal of 53 is One Quarter of Maximal
* Probability of no 2 People out of 53 Sharing the Same Birthday
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:53
53,955

== Number ==
$53 \, 955$ (fifty-three thousand, nine hundred and fifty-five) is:
:$3^2 \times 5 \times 11 \times 109$
:One of the elements the $2$-cycle to which Kaprekar's process on some $5$-digit numbers converges:
::$53 \, 955 \to 59 \, 994 \to 53 \, 955$
== Also see ==
* 
Category:Specific Numbers
Category:53,955
530

== Number ==
$530$ (five hundred and thirty) is:
:$2 \times 5 \times 53$
:The $42$nd untouchable number after $2$, $5$, $52$, $88$, $96$, $120$, $124$, $\ldots$, $406$, $408$, $426$, $430$, $448$, $472$, $474$, $498$, $516$, $518$, $520$
== Also see ==
* 
Category:Specific Numbers
Category:530
531,441

== Number ==
$531 \, 441$ (five hundred and thirty-one thousand, four hundred and forty-one) is:
:$3^{12}$
:The $6$th power of $9$ after $(1)$, $9$, $81$, $729$, $6561$, $59 \, 049$:
::$531 \, 441 = 9^6$
:The $9$th $6$th power after $1$, $64$, $729$, $4096$, $15 \, 625$, $46 \, 656$, $117 \, 649$, $262 \, 144$:
::$531 \, 441 = 9 \times 9 \times 9 \times 9 \times 9 \times 9$
:The $12$th power of $3$ after $(1)$, $3$, $9$, $27$, $81$, $243$, $729$, $2187$, $6561$, $19 \, 683$, $59 \, 049$, $177 \, 147$:
::$531 \, 441 = 3^{12}$
:The $81$st cube number:
::$531 \, 441 = 81 \times 81 \times 81$
:The $729$th square number:
::$531 \, 441 = 729 \times 729$
== Also see ==
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:531,441
532

== Number ==
$532$ (five hundred and thirty-two) is:
:$2^2 \times 7 \times 19$
:The $19$th pentagonal number after $1$, $5$, $12$, $22$, $35$, $51$, $70$, $92$, $117$, $145$, $176$, $210$, $247$, $287$, $330$, $330$, $376$, $425$, $477$:
::$532 = \ds \sum_{k \mathop = 1}^{19} \paren {3 k - 2} = \dfrac {19 \paren {3 \times 19 - 1} } 2$
:The $37$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $392$, $425$, $442$, $477$, $495$:
::$532 = \ds \sum_{k \mathop = 1}^{19} \paren {3 k - 2} = \dfrac {19 \paren {3 \times 19 - 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:532
5329

== Number ==
$5329$ (five thousand, three hundred and twenty-nine) is:
:$73^2$
:The $73$rd square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $4096$, $4225$, $4356$, $4489$, $4624$, $4761$, $4900$, $5041$, $5184$
::$5329 = 73 \times 73$
== Also see ==
* 
Category:Specific Numbers
Category:5329
5335

== Number ==
$5335$ (five thousand, three hundred and thirty-five) is:
:$5 \times 11 \times 97$
:The magic constant of a magic square of order $22$, after $1$, $(5)$, $15$, $34$, $\ldots$, $870$, $1105$, $1379$, $1695$, $2056$, $2465$, $2925$, $3439$, $4010$, $4641$:
::$\ds 5335 = \dfrac 1 {22} \sum_{k \mathop = 1}^{22^2} k = \dfrac {22 \paren {22^2 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:5335
534,494,836

== Number ==
$534 \, 494 \, 836$ is:
:$2^2 \times 419 \times 318 \, 911$
:The $30$th pluperfect digital invariant after $1$, $2$, $3$, $4$, $5$, $6$, $\ldots$, $88 \, 593 \, 477$, $146 \, 511 \, 208$, $472 \, 335 \, 975$:
== Also see ==
* 
Category:Specific Numbers
Category:534,494,836
5347

== Number ==
$5347$ (five thousand, three hundred and forty-seven) is:
:The $707$th prime number
:The $3$rd after $587$, $1777$ of a sequence of $11$ primes formed by tripling and adding $16$:
::$5347 = 3 \times 1777 + 16$
== Also see ==
* 
* 
Category:Specific Numbers
Category:5347
535

== Number ==
$535$ (five hundred and thirty-five) is:
:$5 \times 107$
:The $13$th palindromic lucky number:
::$1$, $3$, $7$, $9$, $33$, $99$, $111$, $141$, $151$, $171$, $303$, $393$, $535$, $\ldots$
:The $23$rd Smith number after $4$, $22$, $27$, $58$, $\ldots$, $346$, $355$, $378$, $382$, $391$, $438$, $454$, $483$, $517$, $526$:
::$5 + 3 + 5 = 5 + 1 + 0 + 7 = 13$
== Also see ==
* 
* 
Category:Specific Numbers
Category:535
5355

== Number ==
$5355$ (five thousand, three hundred and fifty-five) is:
:$3^2 \times 5 \times 7 \times 17$
:The $8$th odd abundant number after $945$, $1575$, $2205$, $2835$, $3465$, $4095$, $4725$:
::$\map {\sigma_1} {5355} - 5355 = 5877 > 5355$
== Also see ==
* 
Category:Specific Numbers
Category:5355
5356

== Number ==
$5356$ (five thousand, three hundred and fifty-six) is:
:$2^2 \times 13 \times 103$
:The $52$nd hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $4005$, $4186$, $4371$, $4560$, $4950$, $5151$:
::$5356 = \ds \sum_{k \mathop = 1}^{52} \paren {4 k - 3} = 52 \paren {2 \times 52 - 1}$
:The $103$rd triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $4560$, $4656$, $4753$, $4851$, $4950$, $5050$, $5151$, $5253$:
::$5356 = \ds \sum_{k \mathop = 1}^{103} k = \dfrac {103 \times \paren {103 + 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:5356
536,870,911

== Number ==
$536 \, 870 \, 911$ (five hundred and thirty-six million, eight hundred and seventy thousand, nine hundred and eleven) is:
:$233 \times 1103 \times 2089$
:The $3$rd Mersenne number after $2047$, $8 \, 388 \, 607$ which is composite:
::$536 \, 870 \, 911 = 2^{29} - 1 = 233 \times 1103 \times 2089$
:The $10$th Mersenne number after $3$, $7$, $31$, $127$, $2047$, $8191$, $131 \, 071$, $524 \, 287$, $8 \, 388 \, 607$:
::$536 \, 870 \, 911 = 2^{29} - 1$
== Historical Note ==
== Also see ==
* 
* 
Category:Specific Numbers
Category:536,870,911
536,870,911/Historical Note

== Historical Note on $536 \, 870 \, 911$ ==
$536 \, 870 \, 911$ was the $4$th Mersenne number that was demonstrated to be composite.
This was done in $1732$ by , who demonstrated it had the divisors $233$ and $1103$.
It is in fact the $3$rd Mersenne number in sequence to be composite.
However, the status of the $4$th such composite Mersenne number, $137 \, 438 \, 953 \, 471$, had been established by in $1640$.
== Sources ==
* : Preface
Category:Historical Notes
536,870,912

== Number ==
$536 \, 870 \, 912$ (five hundred and thirty-six million, eight hundred and seventy thousand, nine hundred and twelve) is:
:$2^{29}$
:The $29$th power of $2$:
::$536 \, 870 \, 912 = 2^{29}$
:The $30$th almost perfect number:
::$\map {\sigma_1} {536 \, 870 \, 912} = 1 \, 073 \, 741 \, 823 = 2 \times 536 \, 870 \, 912 - 1$
:Of particular interest, as it is almost pandigital: only $4$ is missing.
== Also see ==
* 
* 
Category:Specific Numbers
Category:536,870,912
537,824

== Number ==
$537 \, 824$ (five hundred and thirty-seven thousand, eight hundred and twenty-four) is:
:$2^5 \times 7^5$
:The $5$th power of $14$ after $(1)$, $14$, $196$, $2744$, $38 \, 416$:
::$537 \, 824 = 14^5$
:The $14$th fifth power after $1$, $32$, $243$, $1024$, $3125$, $7776$, $16 \, 807$, $32 \, 768$, $59 \, 049$, $100 \, 000$, $161 \, 051$, $248 \, 832$, $371 \, 293$:
::$537 \, 824 = 14 \times 14 \times 14 \times 14 \times 14$
== Also see ==
* 
* 
Category:Specific Numbers
Category:537,824
538

== Number ==
$538$ (five hundred and thirty-eight) is:
:$2 \times 269$
:The $3$rd positive integer after $38$, $462$ whose square ends in $444$:
::$538^2 = \left({500 + 38}\right)^2 = \left({250 + 38}\right) \times 1000 + 1444 = 289 \, 144$
== Also see ==
* 
Category:Specific Numbers
Category:538
54

== Number ==
$54$ (fifty-four) is:
:$2 \times 3^3$
:The $2$nd integer solution to $\map {\sigma_1} n = \map {\sigma_1} {n + 2}$ after $33$:
::$\map {\sigma_1} {54} = 120 = \map {\sigma_1} {56}$
:The $10$th positive integer $n$ after $5$, $11$, $17$, $23$, $29$, $30$, $36$, $42$, $48$ such that no factorial of an integer can end with $n$ zeroes.
:The $12$th semiperfect number after $6$, $12$, $18$, $20$, $24$, $28$, $30$, $36$, $40$, $42$, $48$:
::$54 = 9 + 18 + 27$
:The $32$nd positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $33$, $37$, $38$, $42$, $43$, $44$, $45$, $46$, $49$, $50$ which cannot be expressed as the sum of distinct pentagonal numbers.
:The $35$th (strictly) positive integer after $1$, $2$, $3$, $\ldots$, $37$, $38$, $42$, $43$, $44$, $48$, $49$, $50$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$
== Also see ==
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:54
54,748

== Number ==
$54 \, 748$ (fifty-four thousand, seven hundred and forty-eight) is:
:$2^2 \times 13 \, 687$
:The $17$th pluperfect digital invariant after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $153$, $370$, $371$, $407$, $1634$, $8208$, $9474$:
::$5^5 + 4^5 + 7^5 + 4^5 + 8^5 = 3125 + 1024 + 16 \, 807 + 1024 + 32 \, 768 = 54 \, 748$
== Also see ==
* 
== Sources ==
* : $54,748$
* : $54,748$
Category:Specific Numbers
Category:54,748
540

== Number ==
$540$ (five hundred and forty) is:
:$2^2 \times 3^3 \times 5$
:The $15$th heptagonal number after $1$, $7$, $18$, $34$, $55$, $81$, $112$, $148$, $189$, $235$, $286$, $342$, $403$, $469$:
::$540 = \ds \sum_{k \mathop = 1}^{15} \paren {5 k - 4} = \dfrac {15 \paren {5 \times 15 - 3} } 2$
:The $27$th positive integer after $64$, $96$, $128$, $144$, $\ldots$, $384$, $400$, $416$, $432$, $448$, $480$, $504$, $512$, $528$ with $6$ or more prime factors:
::$540 = 2 \times 2 \times 3 \times 3 \times 3 \times 5$
:The $43$rd untouchable number after $2$, $5$, $52$, $88$, $96$, $120$, $124$, $\ldots$, $406$, $408$, $426$, $430$, $448$, $472$, $474$, $498$, $516$, $518$, $520$, $530$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:540
541

== Number ==
$541$ (five hundred and forty-one) is:
:The $100$th prime number
:The $17$th of $29$ primes of the form $2 x^2 + 29$:
::$2 \times 16^2 + 29 = 541$ ()
:The $38$th long period prime after $7$, $17$, $19$, $23$, $29$, $\ldots$, $383$, $389$, $419$, $433$, $461$, $487$, $491$, $499$, $503$, $509$
== Also see ==
* 
* 
Category:Specific Numbers
Category:541
5419

== Number ==
$5419$ (five thousand, four hundred and nineteen) is:
:The $716$th prime number
:The $43$rd centered hexagonal number after $1$, $7$, $19$, $37$, $61$, $91$, $127$, $\ldots$, $4219$, $4447$, $4681$, $4921$, $5167$:
::$5419 = \ds 1 + \sum_{k \mathop = 1}^{43 - 1} 6 k = 43^3 - 42^3$
== Also see ==
* 
* 
Category:Specific Numbers
Category:5419
5452

== Number ==
$5452$ (five thousand, four hundred and fifty-two) is:
:$2^2 \times 29 \times 47$
:The $47$th heptagonal number after $1$, $7$, $18$, $34$, $55$, $81$, $112$, $\ldots$, $3744$, $3940$, $4141$, $4347$, $4558$, $4774$, $4995$, $5221$:
::$5452 = \ds \sum_{k \mathop = 1}^{47} \paren {5 k - 4} = \dfrac {47 \paren {5 \times 47 - 3} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:5452
5456

== Number ==
$5456$ (five thousand, four hundred and fifty-six) is:
:$2^4 \times 11 \times 31$
:The $31$st tetrahedral number, after $1$, $4$, $10$, $20$, $35$, $\ldots$, $816$, $969$, $1140$, $1330$, $1540$, $1771$, $2024$, $2300$, $2600$, $2925$, $3276$, $3276$, $4060$, $4495$, $4960$:
::$5456 = \ds \sum_{k \mathop = 1}^{31} \frac {k \paren {k + 1} } 2 = \dfrac {31 \paren {31 + 1} \paren {31 + 2} } 6$
== Also see ==
* 
Category:Specific Numbers
Category:5456
546

== Number ==
$546$ (five hundred and forty-six) is:
:$2 \times 3 \times 7 \times 13$
:The $16$th integer $m$ such that $m! - 1$ (its factorial minus $1$) is prime:
::$3$, $4$, $6$, $7$, $12$, $14$, $30$, $32$, $33$, $38$, $94$, $166$, $324$, $379$, $469$, $546$
== Also see ==
* 
Category:Specific Numbers
Category:546
5460

== Number ==
$5460$ (five thousand, four hundred and sixty) is:
:$2^2 \times 3 \times 5 \times 7 \times 13$
:The $1$st of the $7$th pair of triangular numbers whose sum and difference are also both triangular:
::$5460 = T_{104}$, $6786 = T_{116}$, $6786 + 5460 = T_{156}$, $6786 - 5460 = T_{51}$
:The $104$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $4560$, $4656$, $4753$, $4851$, $4950$, $5050$, $5151$, $5253$, $5356$:
::$5460 = \ds \sum_{k \mathop = 1}^{104} k = \dfrac {104 \times \paren {104 + 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:5460
5461

== Number ==
$5461$ (five thousand, four hundred and sixty-one) is:
:$43 \times 127$
:The $17$th Poulet number after $341$, $561$, $645$, $1105$, $1387$, $1729$, $1905$, $2047$, $2465$, $2701$, $2821$, $3277$, $4033$, $4369$, $4371$, $4681$:
::$2^{5461} \equiv 2 \pmod {5461}$: $5461 = 43 \times 127$
:The $43$rd octagonal number, after $1$, $8$, $21$, $40$, $65$, $\ldots$, $3816$, $4033$, $4256$, $4485$, $4720$, $4961$, $5208$:
::$5461 = \ds \sum_{k \mathop = 1}^{43} \paren {6 k - 5} = 43 \paren {3 \times 43 - 2}$
== Also see ==
* 
* 
Category:Specific Numbers
Category:5461
547

== Number ==
$547$ (five hundred and forty-seven) is:
:The $101$st prime number
:The smaller of the $8$th pair of primes whose prime gap is $10$:
::$557 - 547 = 10$
:The $14$th centered hexagonal number after $1$, $7$, $19$, $37$, $61$, $91$, $127$, $169$, $217$, $271$, $331$, $397$, $469$:
::$547 = \ds 1 + \sum_{k \mathop = 1}^{14 - 1} 6 k = 14^3 - 13^3$
:The $35$th left-truncatable prime after $2$, $3$, $5$, $7$, $\ldots$, $197$, $223$, $283$, $313$, $317$, $337$, $347$, $353$, $367$, $373$, $383$, $397$, $443$, $467$, $523$
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:547
5476

== Number ==
$5476$ (five thousand, four hundred and seventy-six) is:
:$2^2 \times 37^2$
:The $74$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $4096$, $4225$, $4356$, $4489$, $4624$, $4761$, $4900$, $5041$, $5184$, $5329$
::$5476 = 74 \times 74$
== Also see ==
* 
Category:Specific Numbers
Category:5476
5477

== Number ==
$5477$ (five thousand, four hundred and seventy-seven) is:
:The $723$rd prime number
:The $16$th prime number of the form $n^2 + 1$ after $2$, $5$, $17$, $37$, $101$, $197$, $257$, $401$, $577$, $677$, $1297$, $1601$, $2917$, $3137$, $4357$:
::$5477 = 74^2 + 1$
== Also see ==
* 
* 
Category:Specific Numbers
Category:5477
548,834

== Number ==
$548 \, 834$ (five hundred and forty-eight thousand, eight hundred and thirty-four) is:
:$2 \times 11 \times 13 \times 19 \times 101$
:The $20$th pluperfect digital invariant after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $153$, $370$, $371$, $407$, $1634$, $8208$, $9474$, $54 \, 748$, $92 \, 727$, $93 \, 084$:
::$5^6 + 4^6 + 8^6 + 8^6 + 3^6 + 4^6 = 15 \, 625 + 4096 + 262 \, 144 + 262 \, 144 + 729 + 4096 = 548 \, 834$
== Also see ==
* 
== Sources ==
* : $548,834$
* : $548,834$
Category:Specific Numbers
Category:548,834
55

== Number ==
$55$ (fifty-five) is:
:$5 \times 11$
:The $1$st of the $3$ repdigit numbers which are also triangular
:The $1$st of the $4$ cubic recurring digital invariants:
::$55 \to 250 \to 133 \to 55$
:The $2$nd square pyramorphic number after $1$:
::$55 = \ds \sum_{k \mathop = 1}^5 k^2 = \dfrac {5 \paren {5 + 1} \paren {2 \times 5 + 1} } 6$
:The $2$nd after $1$ of the $4$ square pyramidal numbers which are also triangular
:The $4$th Kaprekar number after $1$, $9$, $45$:
::$55^2 = 3025 \to 30 + 25 = 55$
:The $5$th heptagonal number after $1$, $7$, $18$, $34$:
::$55 = 1 + 7 + 11 + 16 + 21 = \dfrac {5 \paren {5 \times 5 - 3} } 2$
:The $5$th square pyramidal number after $1$, $5$, $14$, $30$:
::$55 = 1 + 4 + 9 + 14 + 25 = \dfrac {5 \paren {5 + 1} \paren {2 \times 5 + 1} } 6$
:The $5$th and last after $0$, $1$, $3$, $21$ of the $5$ Fibonacci numbers which are also triangular
:The $5$th palindromic triangular number after $0$, $1$, $3$, $6$
::$55 = \dfrac {10 \times \paren {10 + 1} } 2$
:The $1$st of those which can be split into two palindromic halves
:The $10$th Fibonacci number, after $1$, $1$, $2$, $3$, $5$, $8$, $13$, $21$, $34$:
::$55 = 21 + 34$
:The $10$th triangular number after $1$, $3$, $6$, $10$, $15$, $21$, $28$, $36$, $45$:
::$55 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = \dfrac {10 \times \paren {10 + 1} } 2$
:The $19$th semiprime after $4$, $6$, $9$, $10$, $14$, $15$, $21$, $22$, $25$, $26$, $33$, $34$, $35$, $38$, $39$, $46$, $49$, $51$:
::$55 = 5 \times 11$
:The $21$st of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:
::$3$, $4$, $9$, $10$, $12$, $16$, $17$, $22$, $23$, $25$, $27$, $29$, $30$, $36$, $38$, $40$, $43$, $48$, $51$, $53$, $55$, $\ldots$
:The $24$th and last after $1$, $2$, $4$, $5$, $6$, $8$, $9$, $12$, $13$, $15$, $16$, $17$, $20$, $24$, $25$, $27$, $28$, $32$, $35$, $26$, $39$, $48$, $51$ of the $24$ positive integers which cannot be expressed as the sum of distinct non-pythagorean primes
:The $26$th odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
::$1$, $3$, $5$, $7$, $\ldots$, $35$, $37$, $41$, $43$, $45$, $47$, $49$, $53$, $55$, $\ldots$
:The $33$rd positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $33$, $37$, $38$, $42$, $43$, $44$, $45$, $46$, $49$, $50$, $54$ which cannot be expressed as the sum of distinct pentagonal numbers
:The $36$th (strictly) positive integer after $1$, $2$, $3$, $\ldots$, $37$, $38$, $42$, $43$, $44$, $48$, $49$, $50$, $54$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$
:There are exactly $55$ sets of $4$ integers $\set {a, b, c, d}$ such that all integers can be written in the form:
::$n = a x^2 + b y^2 + c z^2 + d w^2$
:for integer $x, y, z, w$
== Also see ==
* Sets of $4$ Integers $a, b, c, d$ for which Every Integer is in form $a x^2 + b y^2 + c z^2 + d u^2$
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== Sources ==
* : $55$
* : $55$
Category:Specific Numbers
Category:55
55,225

== Number ==
$55 \, 225$ (fifty-five thousand, two hundred and twenty-five) is:
:$5^2 \times 47^2$
:The $21$st square after $1$, $4$, $9$, $16$, $25$, $\ldots$, $441$, $484$, $676$, $1444$, $7744$, $11 \, 881$, $29 \, 929$, $44 \, 944$ which has no more than $2$ distinct digits and does not end in $0$:
::$55 \, 225 = 235^2$
:The $235$th square number:
::$55 \, 225 = 235 \times 235$
== Also see ==
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Category:Specific Numbers
Category:55,225
550

== Number ==
$550$ (five hundred and fifty) is:
:$2 \times 5^2 \times 11$
:The $9$th primitive abundant number after $20$, $70$, $88$, $104$, $272$, $304$, $368$, $464$:
::$1 + 2 + 5 + 10 + 11 + 22 + 25 + 50 + 55 + 110 + 275 = 566 > 550$
:The $10$th pentagonal pyramidal number after $1$, $6$, $12$, $40$, $75$, $126$, $196$, $288$, $405$:
::$550 = \ds \sum_{k \mathop = 1}^{10} \dfrac {k \paren {3 k - 1} } 2 = \dfrac {10^2 \paren {10 + 1} } 2$
:The $13$th primitive semiperfect number after $6$, $20$, $28$, $88$, $104$, $272$, $304$, $350$, $368$, $464$, $490$, $496$:
::$550 = 2 + 11 + 22 + 25 + 50 + 55 + 110 + 275$
:The smallest positive integer which can be expressed as the sum of $2$ distinct lucky numbers in $26$ different ways
== Also see ==
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Category:Specific Numbers
Category:550