nyuuzyou/wikis huggingface dataset:
63,636

== Number ==
$63 \, 636$ (sixty-three thousand, six hundred and thirty-six) is:
:$2^2 \times 3 \times 5303$
:The $10$th integer after $91$, $428$, $573$, $727$, $846$, $7810$, $9079$, $9901$, $36 \, 365$ whose square has a decimal representation consisting of the concatenation of $2$ consecutive integers:
::$63 \, 636^2 = 4 \, 049 \, 540 \, 496$
:The $7$th integer after $428$, $573$, $727$, $846$, $7810$, $36 \, 365$ whose square has a decimal representation consisting of the concatenation of $2$ consecutive increasing integers:
::$63 \, 636^2 = 4 \, 049 \, 540 \, 496$
== Also see ==
* 
* 
Category:Specific Numbers
Category:63,636
63,954

== Number ==
$63 \, 954$ (sixty-three thousand, nine hundred and fifty-four) is:
:$2 \times 3^2 \times 11 \times 17 \times 19$
:One of the elements of one of the $4$-cycles to which Kaprekar's process on some $5$-digit numbers converges:
::$63 \, 954 \to 61 \, 974 \to 82 \, 962 \to 75 \, 933 \to 63 \, 954$
== Also see ==
* 
Category:Specific Numbers
Category:63,954
630

== Number ==
$630$ (six hundred and thirty) is:
:$2 \times 3^2 \times 5 \times 7$
:The $18$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $120$, $153$, $190$, $231$, $276$, $325$, $378$, $435$, $496$, $561$:
::$630 = \ds \sum_{k \mathop = 1}^{18} \paren {4 k - 3} = 18 \paren {2 \times 18 - 1}$
:The $20$th integer $n$ after $1$, $3$, $15$, $30$, $35$, $56$, $70$, $78$, $105$, $140$, $168$, $190$, $210$, $248$, $264$, $357$, $420$, $570$, $616$ with the property that $\map {\sigma_0} n \divides \map \phi n \divides \map {\sigma_1} n$:
::$\map {\sigma_0} {630} = 24$, $\map \phi {630} = 144$, $\map {\sigma_1} {630} = 1872$
:The $35$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $325$, $351$, $378$, $406$, $435$, $465$, $496$, $528$, $561$, $595$:
::$630 = \ds \sum_{k \mathop = 1}^{35} k = \dfrac {35 \times \paren {35 + 1} } 2$
:The $39$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $\ldots$, $120$, $144$, $168$, $180$, $210$, $216$, $240$, $288$, $300$, $336$, $360$, $420$, $480$, $504$, $600$:
::$\map {\sigma_1} {630} = 1872$
== Also see ==
* 
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Category:Specific Numbers
Category:630
631

== Number ==
$631$ (six hundred and thirty-one) is:
:The $115$th prime number
:The smaller of the $10$th pair of primes whose prime gap is $10$:
::$641 - 631 = 10$
:The $15$th centered hexagonal number after $1$, $7$, $19$, $37$, $61$, $91$, $127$, $169$, $217$, $271$, $331$, $397$, $469$, $547$:
::$631 = \ds 1 + \sum_{k \mathop = 1}^{15 - 1} 6 k = 15^3 - 14^3$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:631
6328

== Number ==
$6328$ (six thousand, three hundred and twenty-nine) is:
:$2^3 \times 7 \times 113$
:The $112$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $5253$, $5356$, $5460$, $5565$, $5671$, $5778$, $5886$, $5995$, $6105$, $6216$:
::$6328 = \ds \sum_{k \mathop = 1}^{112} k = \dfrac {112 \times \paren {112 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:6328
634

== Number ==
$634$ (six hundred and thirty-four) is:
:$2 \times 317$
:The smallest positive integer which can be expressed as the sum of $2$ distinct lucky numbers in $25$ different ways
:The $28$th Smith number after $4$, $22$, $27$, $58$, $\ldots$, $454$, $483$, $517$, $526$, $535$, $562$, $576$, $588$, $627$:
::$6 + 3 + 4 = 2 + 3 + 1 + 7 = 13$
== Also see ==
* 
* 
Category:Specific Numbers
Category:634
6348

== Number ==
$6348$ (six thousand, three hundred and forty-eight) is:
:$2^2 \times 3 \times 23^2$
:The $23$rd pentagonal pyramidal number after $1$, $6$, $12$, $40$, $75$, $\ldots$, $1470$, $1800$, $2176$, $2601$, $3078$, $3610$, $4200$, $4851$, $5566$:
::$6348 = \ds \sum_{k \mathop = 1}^{23} \dfrac {k \paren {3 k - 1} } 2 = \dfrac {23^2 \paren {23 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:6348
635,318,657

== Number ==
$635 \, 318 \, 657$ is:
:$41 \times 113 \times 241 \times 569$
:The smallest positive integer which can be expressed as the sum of $2$ fourth powers in $2$ different ways:
::$635 \, 318 \, 657 = 59^4 + 158^4 = 133^4 + 134^4$
== Also see ==
* Smallest Positive Integer which is Sum of 2 Fourth Powers in 2 Ways
== Sources ==
* : $635,318,657$
* : $635,318,657$
Category:Specific Numbers
Category:635,318,657
636

== Number ==
$636$ (six hundred and thirty-six) is:
:$2^2 \times 3 \times 53$
:The $29$th Smith number after $4$, $22$, $27$, $58$, $\ldots$, $454$, $483$, $517$, $526$, $535$, $562$, $576$, $588$, $627$, $634$:
::$6 + 3 + 6 = 2 + 2 + 3 + 5 + 3 = 15$
== Also see ==
* 
Category:Specific Numbers
Category:636
6368

== Number ==
$6368$ (six thousand, three hundred and sixty-eight) is:
:$2^5 \times 199$
:The larger of the $5$th amicable pair, with $6232$:
::$\map {\sigma_1} {6232} = \map {\sigma_1} {6368} = 12 \, 600 = 6232 + 6368$
== Also see ==
* 
Category:Specific Numbers
Category:6368
637,832,238,736

== Number ==
$637 \, 832 \, 238 \, 736$ is:
:$2^4 \times 7^2 \times 11^2 \times 2593^2$
:The $2$nd palindromic square with an even number of digits with a non-palindromic square root:
::$637 \, 832 \, 238 \, 736 = 798 \, 644^2$
:The $10$th palindromic square with a non-palindromic square root:
::$637 \, 832 \, 238 \, 736 = 798 \, 644^2$
:The $798 \, 644$th square number:
::$637 \, 832 \, 238 \, 736 = 798 \, 644 \times 798 \, 644$
== Also see ==
* 
* 
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== Sources ==
* : $637,832,238,736$
* : $637,832,238,736$
Category:Specific Numbers
Category:637,832,238,736
638,523,249

== Number ==
$638 \, 523 \, 249$ is:
:$3 \times 7 \times 13 \times 107 \times 21 \, 859$
:The $3$rd known positive integer whose fourth power can be expressed as the sum of $3$ fourth powers, thus refuting the Euler Quartic Conjecture:
::$638 \, 523 \, 249^4 = 630 \, 662 \, 624^4 + 275 \, 156 \, 240^4 + 219 \, 076 \, 465^4$
== Also see ==
* 
Category:Specific Numbers
Category:638,523,249
638,817,435,613,190,763,972,389,505,493

== Number ==
$638 \, 817 \, 435 \, 613 \, 190 \, 763 \, 972 \, 389 \, 505 \, 493$ is:
:$6 \, 260 \, 874 \, 567 \, 713 \times 102 \, 033 \, 258 \, 884 \, 875 \, 061$
:The $173$rd Fibonacci number
== Sources ==
* 
Category:Specific Numbers
Category:638,817,435,613,190,763,972,389,505,493
6380

== Number ==
$6380$ (six thousand, three hundred and eighty) is:
:$2^2 \times 5 \times 11 \times 29$
:The $19$th integer $m$ such that $m! + 1$ (its factorial plus $1$) is prime:
::$0$, $1$, $2$, $3$, $11$, $27$, $37$, $41$, $73$, $77$, $116$, $154$, $320$, $340$, $399$, $427$, $872$, $1477$, $6380$
== Also see ==
* 
Category:Specific Numbers
Category:6380
64

== Number ==
$64$ (sixty-four) is:
:$2^6$
:Represented as $100$ in octal and $1 \, 000 \, 000$ in binary.
:The $1$st positive integer with $6$ or more prime factors:
::$64 = 2 \times 2 \times 2 \times 2 \times 2 \times 2$
:The $4$th of the $1$st ordered quadruple of consecutive integers that have divisor sums which are strictly increasing:
::$\map {\sigma_1} {61} = 62$, $\map {\sigma_1} {62} = 96$, $\map {\sigma_1} {63} = 104$, $\map {\sigma_1} {64} = 127$
:The $2$nd $6$th power after $1$:
::$64 = 2 \times 2 \times 2 \times 2 \times 2 \times 2$
:The $2$nd power of $8$ after $(1)$, $8$:
::$64 = 8^2$
:The $2$nd element of the $3$rd pair of integers $m$ whose values of $m \map {\sigma_0} m$ is equal:
::$56 \times \map {\sigma_0} {56} = 448 = 64 \times \map {\sigma_0} {64}$
:The $3$rd power of $4$ after $(1)$, $4$, $16$:
::$64 = 4^3$
:The $4$th cube number after $1$, $8$, $27$:
::$64 = 4 \times 4 \times 4$
:The total number of permutations of $r$ objects from a set of $4$ objects, where $1 \le r \le 4$
:The $6$th power of $2$ after $(1)$, $2$, $4$, $8$, $16$, $32$:
::$64 = 2^6$
:The $6$th positive integer which cannot be expressed as the sum of a square and a prime:
::$1$, $10$, $25$, $34$, $58$, $64$, $\ldots$
:The $7$th almost perfect number after $1$, $2$, $4$, $8$, $16$, $32$:
::$\map {\sigma_1} {64} = 127 = 2 \times 64 - 1$
:The $8$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $49$:
::$64 = 8 \times 8$
:The $8$th square after $1$, $4$, $9$, $16$, $25$, $36$, $49$ which has no more than $2$ distinct digits and does not end in $0$:
::$64 = 8^2$
:The $11$th powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $27$, $32$, $36$, $49$
:The $24$th of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:
::$3$, $4$, $9$, $10$, $12$, $16$, $17$, $22$, $23$, $25$, $27$, $29$, $30$, $36$, $38$, $40$, $43$, $48$, $51$, $53$, $55$, $61$, $62$, $64$, $\ldots$
== Also see ==
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== Sources ==
* : $64$
* : $64$
Category:Specific Numbers
Category:64
64,000

== Number ==
$64 \, 000$ (sixty-four thousand) is:
:$2^9 \times 5^3$
:The $3$rd (and last) after $1$, $21 \, 952$ of the $3$ positive integers whose divisor count equals its cube root:
::$\map {\sigma_0} {64 \, 000} = 40$
:The $40$th cube number:
::$64 \, 000 = 40 \times 40 \times 40$
== Also see ==
* 
* 
== Sources ==
* : $64,000$
Category:Specific Numbers
Category:64,000
64,795,852,800

== Number ==
$64 \, 795 \, 852 \, 800$ is:
:$2^{14} \times 3^6 \times 5^2 \times 7 \times 31$
:The sum of $3$ distinct amicable pairs:
::$29 \, 912 \, 035 \, 725$ and $34 \, 883 \, 817 \, 075$
::$31 \, 695 \, 652 \, 275$ and $33 \, 100 \, 200 \, 525$
::$32 \, 129 \, 958 \, 525$ and $32 \, 665 \, 894 \, 275$
== Also see ==
* Common Sum of 3 Distinct Amicable Pairs
== Sources ==
* : $64,795,852,800$
Category:Specific Numbers
Category:64,795,852,800
640

== Number ==
$640$ (six hundred and forty) is:
:$2^7 \times 5$
:The $12$th positive integer after $128$, $192$, $256$, $288$, $320$, $384$, $432$, $448$, $480$, $512$, $576$ with $7$ or more prime factors:
::$640 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \paren {\times \, 5}$
== Also see ==
* 
Category:Specific Numbers
Category:640
6400

== Number ==
$6400$ (six thousand, four hundred) is:
:$2^8 \times 5^2$
:The $31$st square number after $1$, $4$, $36$, $121$, $144$, $256$, $\ldots$, $5184$, $5776$, $6084$ to be the divisor sum value of some (strictly) positive integer:
::$6400 = \map {\sigma_1} {3591} = \map {\sigma_1} {4179} = \map {\sigma_1} {4503} = \map {\sigma_1} {6169}$
:The $80$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $5041$, $5184$, $5329$, $5476$, $5625$, $5776$, $5929$, $6084$, $6241$
::$6400 = 80 \times 80$
== Also see ==
* 
* 
Category:Specific Numbers
Category:6400
641

== Number ==
$641$ (six hundred and forty-one) is:
:The $116$th prime number
:The smaller of the two prime factors of the $5$th Fermat number $2^{\paren {2^5} } + 1$, the other being $6 \, 700 \, 417$:
::$641 = 5 \times 2^7 + 1$
:The larger of the $10$th pair of primes whose prime gap is $10$:
::$641 - 631 = 10$
:The $14$th Proth prime after $3$, $5$, $13$, $17$, $41$, $97$, $113$, $193$, $241$, $257$, $353$, $449$, $577$:
::$641 = 10 \times 2^6 + 1$
:The $28$th Sophie Germain prime after $2$, $3$, $5$, $11$, $23$, $29$, $41$, $\ldots$, $233$, $239$, $251$, $281$, $293$, $359$, $419$, $431$, $443$, $491$, $509$, $593$:
::$2 \times 641 + 1 = 1283$, which is prime.
:The smaller of the $29$th pair of twin primes:
::$643 - 641 = 2$
== Also see ==
* 
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* 
* 
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== Sources ==
* : $257$
* : $641$
* : $257$
* : $641$
Category:Specific Numbers
Category:641
6426

== Number ==
$6426$ (six thousand, four hundred and twenty-six) is:
:$2 \times 3^3 \times 7 \times 17$
:The $51$st heptagonal number after $1$, $7$, $18$, $34$, $55$, $81$, $112$, $\ldots$, $3744$, $3940$, $4141$, $4347$, $4558$, $4774$, $4995$, $5221$, $5452$, $5688$, $5929$, $6175$:
::$6426 = \ds \sum_{k \mathop = 1}^{51} \paren {5 k - 4} = \dfrac {51 \paren {5 \times 51 - 3} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:6426
643

== Number ==
$643$ (six hundred and forty-three) is:
:The $117$th prime number
:The larger of the $29$th pair of twin primes:
::$643 - 641 = 2$
:The $38$th left-truncatable prime after $2$, $3$, $5$, $7$, $\ldots$, $443$, $467$, $523$, $547$, $613$, $617$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:643
6435

== Number ==
$6435$ (six thousand, four hundred and thirty-five) is:
:$3^2 \times 5 \times 7 \times 19$
:The $11$th odd abundant number after $945$, $1575$, $2205$, $2835$, $3465$, $4095$, $4725$, $5355$, $5775$, $5985$:
::$\map {\sigma_1} {6435} - 6435 = 6669 > 6435$
== Also see ==
* 
Category:Specific Numbers
Category:6435
6441

== Number ==
$6441$ (six thousand, four hundred and forty-one) is:
:$3 \times 19 \times 113$
:The $57$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $4005$, $4186$, $4371$, $4560$, $4950$, $5151$, $5356$, $5565$, $5778$, $5995$, $6216$:
::$6441 = \ds \sum_{k \mathop = 1}^{57} \paren {4 k - 3} = 57 \paren {2 \times 57 - 1}$
:The $113$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $5253$, $5356$, $5460$, $5565$, $5671$, $5778$, $5886$, $5995$, $6105$, $6216$, $6328$:
::$6441 = \ds \sum_{k \mathop = 1}^{113} k = \dfrac {113 \times \paren {113 + 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:6441
645

== Number ==
$645$ (six hundred and forty-five) is:
:$3 \times 5 \times 43$
:The $3$rd Poulet number after $341$, $561$:
::$2^{645} \equiv 2 \pmod {645}$: $645 = 3 \times 5 \times 43$
:The $8$th Fermat pseudoprime to base $4$ after $15$, $85$, $91$, $341$, $435$, $451$, $561$:
::$4^{645} \equiv 4 \pmod {645}$
:The $9$th heptagonal pyramidal number after $1$, $8$, $26$, $60$, $115$, $196$, $308$, $456$:
::$645 = 1 + 7 + 18 + 34 + 55 + 81 + 112 + 148 + 189 = \dfrac {9 \paren {9 + 1} \paren {5 \times 9 - 2} } 6$
:The $15$th octagonal number, after $1$, $8$, $21$, $40$, $65$, $96$, $133$, $176$, $225$, $280$, $341$, $408$, $481$, $560$:
::$645 = \ds \sum_{k \mathop = 1}^{15} \paren {6 k - 7} = 15 \paren {3 \times 15 - 2}$
:The $30$th Smith number after $4$, $22$, $27$, $58$, $\ldots$, $454$, $483$, $517$, $526$, $535$, $562$, $576$, $588$, $627$, $634$, $636$:
::$6 + 4 + 5 = 3 + 5 + 4 + 3 = 15$
== Also see ==
* 
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== Sources ==
* : $645$
* : $645$
Category:Specific Numbers
Category:645
6469

== Number ==
$6469$ (six thousand, four hundred and sixty-nine) is:
:The $839$th prime number
:The $15$th minimal prime base $10$ after $2$, $3$, $5$, $7$, $11$, $19$, $41$, $61$, $89$, $409$, $449$, $499$, $881$, $991$
== Also see ==
* 
* 
Category:Specific Numbers
Category:6469
647

== Number ==
$647$ (six hundred and forty-seven) is:
:The $118$th prime number
:The $39$th left-truncatable prime after $2$, $3$, $5$, $7$, $\ldots$, $443$, $467$, $523$, $547$, $613$, $617$, $643$
:The $43$rd long period prime after $7$, $17$, $19$, $23$, $29$, $\ldots$, $491$, $499$, $503$, $509$, $541$, $571$, $577$, $593$, $619$
== Also see ==
* 
* 
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Category:Specific Numbers
Category:647
648

== Number ==
$648$ (six hundred and forty-eight) is:
:$2^3 \times 3^4$
:The $43$rd powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $\ldots$, $400$, $432$, $441$, $484$, $500$, $512$, $529$, $576$, $625$:
::$648 = 2^3 \times 3^4$
:The $13$th positive integer after $128$, $192$, $256$, $288$, $320$, $384$, $432$, $448$, $480$, $512$, $576$, $640$ with $7$ or more prime factors:
::$648 = 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3$
:The $31$st Smith number after $4$, $22$, $27$, $58$, $\ldots$, $454$, $483$, $517$, $526$, $535$, $562$, $576$, $588$, $627$, $634$, $636$, $645$:
::$6 + 4 + 8 = 2 + 2 + 2 + 3 + 3 + 3 + 3 = 18$
== Also see ==
* 
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Category:Specific Numbers
Category:648
65

== Number ==
$65$ (sixty-five) is:
:$5 \times 13$
:The $1$st rare number:
::$65 + 56 = 11^2$, $65 - 56 = 3^2$
:The $2$nd positive integer after $50$ which can be expressed as the sum of two square numbers in two or more different ways:
::$65 = 8^2 + 1^2 = 7^2 + 4^2$
:and the smallest such which is also the sum of $2$ cube numbers:
::$65 = 8^2 + 1^2 = 7^2 + 4^2 = 1^3 + 4^3$
:The $3$rd inconsummate number after $62$, $63$:
::$\nexists n \in \Z_{>0}: n = 65 \times \map {s_{10} } n$
:The $5$th octagonal number, after $1$, $8$, $21$, $40$:
::$65 = 1 + 7 + 13 + 19 + 25 = 5 \paren {3 \times 5 - 2}$
:The $5$th Cullen number after $1$, $3$, $9$, $25$:
::$65 = 4 \times 2^4 + 1$
:The magic constant of a magic square of order $5$, after $1$, $(5)$, $15$, $34$:
::$65 = \ds \dfrac 1 5 \sum_{k \mathop = 1}^{5^2} k = \dfrac {5 \paren {5^2 + 1} } 2$
:The $8$th integer after $7$, $13$, $19$, $35$, $38$, $41$, $57$ the decimal representation of whose square can be split into two parts which are each themselves square:
::$65^2 = 4225$; $4 = 2^2$, $225 = 15^2$
:The $9$th positive integer $n$ after $0$, $1$, $5$, $25$, $29$, $41$, $49$, $61$ such that the Fibonacci number $F_n$ ends in $n$
:The $23$rd semiprime:
::$65 = 5 \times 13$
:The $37$th positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $44$, $45$, $46$, $49$, $50$, $54$, $55$, $59$, $60$, $61$ which cannot be expressed as the sum of distinct pentagonal numbers.
:The $39$th (strictly) positive integer after $1$, $2$, $3$, $\ldots$, $37$, $38$, $42$, $43$, $44$, $48$, $49$, $50$, $54$, $55$, $60$, $61$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$
:In the smallest equilateral triangle with sides of integer length ($112$) which contains a point which is an integer distance from each vertex, the distance from that point to its middle vertex (the other two being $57$ and $73$).
== Also see ==
* Smallest Equilateral Triangle with Internal Point at Integer Distances from Vertices
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== Sources ==
* : $65$
* : $65$
Category:Specific Numbers
Category:65
65,536

== Number ==
$65 \, 536$ (sixty-five thousand, five hundred and thirty-six) is:
:$2^{16}$
:The only known power of $2$ whose digits do not contain $1$, $2$, $4$ or $8$.
:The actual number of bytes in $64 \, \mathrm{KB}$ of computer memory.
:The $4$th eighth power after $1$, $256$, $6561$:
::$65 \, 536 = 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4$
:The $4$th power of $16$ after $(1)$, $16$, $256$, $4096$:
::$65 \, 536 = 16^4$
:The $8$th power of $4$ after $(1)$, $4$, $16$, $64$, $256$, $1024$, $4096$, $16 \, 384$:
::$65 \, 536 = 4^8$
:The $16$th fourth power after $1$, $16$, $81$, $256$, $625$, $1296$, $2401$, $4096$, $6561$, $10 \, 000$, $14 \, 641$, $20 \, 736$, $28 \, 561$, $38 \, 416$, $50 \, 625$:
::$65 \, 536 = 16 \times 16 \times 16 \times 16$
:The $16$th power of $2$ after $(1)$, $2$, $4$, $8$, $16$, $32$, $64$, $128$, $256$, $512$, $1024$, $2048$, $4096$, $8192$, $16 \, 384$, $32 \, 768$:
::$65 \, 536 = 2^{16}$
:The $17$th almost perfect number after $1$, $2$, $4$, $8$, $16$, $32$, $64$, $128$, $256$, $512$, $1024$, $2048$, $4096$, $8192$, $16 \, 384$, $32 \, 768$:
::$\map {\sigma_1} {65 \, 536} = 131 \, 071 = 2 \times 65 \, 536 - 1$
:The $256$th square number:
::$65 \, 536 = 256 \times 256$
== Also see ==
* Powers of 2 not containing Digit Power of 2
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== Sources ==
* : $65,536$
* : $65,536$
Category:Specific Numbers
Category:65,536
65,537

== Number ==
$65 \,537$ (sixty-five thousand, five hundred and thirty-seven) is:
:A prime number
:The $5$th Fermat number and Fermat prime after $3$, $5$, $17$, $257$:
::$65 \,537 = 2^{\paren {2^4} } + 1 = 2^{16} + 1$
== Also see ==
* 
* 
== Sources ==
* : $65,537$
* : $65,537$
Category:Specific Numbers
Category:65,537
650

== Number ==
$650$ (six hundred and fifty) is:
:$2 \times 5^2 \times 13$
:The $3$rd positive integer after $325$, $425$ which can be expressed as the sum of two square numbers in $3$ distinct ways:
::$650 = 25^2 + 5^2 = 23^2 + 11^2 = 19^2 + 17^2$
:The $11$th primitive abundant number after $20$, $70$, $88$, $104$, $272$, $304$, $368$, $464$, $550$, $572$:
::$1 + 2 + 5 + 10 + 13 + 25 + 26 + 50 + 65 + 130 + 325 = 652 > 650$
:The $12$th square pyramidal number after $1$, $5$, $14$, $30$, $55$, $91$, $140$, $204$, $285$, $385$, $506$:
::$650 = \ds \sum_{k \mathop = 1}^{12} k^2 = \dfrac {12 \paren {12 + 1} \paren {2 \times 12 + 1} } 6$
:The $15$th primitive semiperfect number after $6$, $20$, $28$, $88$, $104$, $272$, $304$, $350$, $368$, $464$, $490$, $496$, $550$, $572$:
::$650 = 1 + 5 + 10 + 13 + 25 + 26 + 50 + 65 + 130 + 325$
:The $24$th integer after $7$, $13$, $19$, $35$, $38$, $\ldots$, $223$, $253$, $285$, $305$, $350$, $380$, $410$, $475$, $487$, $570$ the decimal representation of whose square can be split into two parts which are each themselves square:
::$650^2 = 422 \, 500$; $4 = 2^2$, $22 \, 500 = 150^2$
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Category:Specific Numbers
Category:650
651

== Number ==
$651$ (six hundred and fifty-one) is:
:$3 \times 7 \times 31$
:The $2$nd positive integer after $353$ such that its fourth power is the sum of $4$ other fourth powers of positive integers with no common factors:
::$651^4 = 240^4 + 340^4 + 430^4 + 599^4$
:The magic constant of a magic cube of order $6$, after $1$, $(9)$, $42$, $130$, $315$:
::$651 = \ds \dfrac 1 {6^2} \sum_{k \mathop = 1}^{6^3} k = \dfrac {6 \paren {6^3 + 1} } 2$
:The $21$st pentagonal number after $1$, $5$, $12$, $22$, $35$, $51$, $70$, $92$, $117$, $145$, $176$, $210$, $247$, $287$, $330$, $330$, $376$, $425$, $477$, $532$, $590$:
::$651 = \ds \sum_{k \mathop = 1}^{21} \paren {3 k - 2} = \dfrac {21 \paren {3 \times 21 - 1} } 2$
:The $31$st number whose divisor sum is square:
::$\map {\sigma_1} {651} = 1024 = 32^2$
:The $41$st generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $392$, $425$, $442$, $477$, $495$, $532$, $551$, $590$, $610$:
::$651 = \ds \sum_{k \mathop = 1}^{21} \paren {3 k - 2} = \dfrac {21 \paren {3 \times 21 - 1} } 2$
:The product with its reversal equals the product of another $3$-digit number with its reversal:
::$651 \times 156 = 372 \times 273$
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* : $651$
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Category:Specific Numbers
Category:651
653

== Number ==
$653$ (six hundred and fifty-three) is:
:The $119$th prime number
:The $12$th balanced prime after $5$, $53$, $157$, $173$, $211$, $257$, $263$, $373$, $563$, $593$, $607$:
::$653 = \dfrac {647 + 659} 2$
:The $29$th Sophie Germain prime after $2$, $3$, $5$, $11$, $23$, $29$, $41$, $\ldots$, $233$, $239$, $251$, $281$, $293$, $359$, $419$, $431$, $443$, $491$, $509$, $593$, $641$:
::$2 \times 653 + 1 = 1307$, which is prime.
:The $40$th left-truncatable prime after $2$, $3$, $5$, $7$, $\ldots$, $443$, $467$, $523$, $547$, $613$, $617$, $643$, $647$
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Category:Specific Numbers
Category:653
6533

== Number ==
$6533$ (six thousand, five hundred and thirty-three) is:
:$47 \times 139$
:The $47$th octagonal number, after $1$, $8$, $21$, $40$, $65$, $\ldots$, $3816$, $4033$, $4256$, $4485$, $4720$, $4961$, $5208$, $5461$, $5720$, $5985$, $6256$:
::$6533 = \ds \sum_{k \mathop = 1}^{47} \paren {6 k - 5} = 47 \paren {3 \times 47 - 2}$
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Category:Specific Numbers
Category:6533
654

== Number ==
$654$ (six hundred and fifty-four) is:
:$2 \times 3 \times 109$
:The $32$nd Smith number after $4$, $22$, $27$, $58$, $\ldots$, $576$, $588$, $627$, $634$, $636$, $645$, $648$:
::$6 + 5 + 4 = 2 + 3 + 1 + 0 + 9 = 15$
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Category:Specific Numbers
Category:654
6545

== Number ==
$6545$ (six thousand, five hundred and forty-five) is:
:$5 \times 7 \times 11 \times 17$
:The $33$rd tetrahedral number, after $1$, $4$, $10$, $20$, $35$, $\ldots$, $3276$, $3276$, $4060$, $4495$, $4960$, $5456$, $5984$:
::$6545 = \ds \sum_{k \mathop = 1}^{33} \frac {k \paren {k + 1} } 2 = \dfrac {33 \paren {33 + 1} \paren {33 + 2} } 6$
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Category:Specific Numbers
Category:6545
6555

== Number ==
$6555$ (six thousand, five hundred and fifty-five) is:
:$3 \times 5 \times 19 \times 23$
:The $114$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $5253$, $5356$, $5460$, $5565$, $5671$, $5778$, $5886$, $5995$, $6105$, $6216$, $6328$, $6441$:
::$6555 = \ds \sum_{k \mathop = 1}^{114} k = \dfrac {114 \times \paren {114 + 1} } 2$
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Category:Specific Numbers
Category:6555