nyuuzyou/wikis huggingface dataset:
449

== Number ==
$449$ (four hundred and forty-nine) is:
:The $87$th prime number
:The smaller of the $5$th pair of primes whose prime gap is $8$:
::$457 - 449 = 8$
:The $11$th minimal prime base $10$ after $2$, $3$, $5$, $7$, $11$, $19$, $41$, $61$, $89$, $409$
:The $12$th Proth prime after $3$, $5$, $13$, $17$, $41$, $97$, $113$, $193$, $241$, $257$, $353$:
::$449 = 7 \times 2^6 + 1$
:The $23$rd near-repdigit prime after $101$, $113$, $131$, $151$, $181$, $191$, $199$, $211$, $223$, $227$, $229$, $233$, $277$, $311$, $313$, $331$, $337$, $353$, $373$, $383$, $433$, $443$
== Also see ==
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:449
4495

== Number ==
$4495$ (four thousand, four hundred and ninety-five) is:
:$5 \times 29 \times 31$
:The $29$th tetrahedral number, after $1$, $4$, $10$, $20$, $35$, $\ldots$, $816$, $969$, $1140$, $1330$, $1540$, $1771$, $2024$, $2300$, $2600$, $2925$, $3276$, $3276$, $4060$:
::$4495 = \ds \sum_{k \mathop = 1}^{29} \frac {k \paren {k + 1} } 2 = \dfrac {29 \paren {29 + 1} \paren {29 + 2} } 6$
== Also see ==
* 
Category:Specific Numbers
Category:4495
45

== Number ==
$45$ (forty-five) is:
:$3^2 \times 5$
:The $2$nd of the $1$st ordered quadruple of consecutive integers that have divisor sums which are strictly decreasing:
::$\map {\sigma_1} {44} = 84$, $\map {\sigma_1} {45} = 78$, $\map {\sigma_1} {46} = 72$, $\map {\sigma_1} {47} = 48$
:The $2$nd of the $1$st pair of consecutive integers which both have $6$ divisors:
::$\map {\sigma_0} {44} = \map {\sigma_0} {45} = 6$
:The $2$nd hexamorphic number after $1$:
::$45 = H_5$
:The $3$rd Kaprekar number after $1$, $9$:
::$45^2 = 2025 \to 20 + 25 = 45$
:The $3$rd Kaprekar triple after $1$, $8$:
::$45^3 = 91 \, 125 \to 9 + 11 + 25 = 45$
:The total of all the entries in a magic square of order $3$, after $1$, $(10)$:
::$45 = \ds \sum_{k \mathop = 1}^{3^2} k = \dfrac {3^2 \paren {3^2 + 1} } 2$
:The $4$th term of the $3$-Göbel sequence after $1$, $2$, $5$:
::$45 = \paren {1 + 1^3 + 2^3 + 5^3} / 3$
:The $5$th hexagonal number after $1$, $6$, $15$, $28$:
::$45 = 1 + 5 + 9 + 13 + 17 = 5 \paren {2 \times 5 - 1}$
:The $5$th positive integer $n$ after $4$, $7$, $15$, $21$ such that $n - 2^k$ is prime for all $k$
:The $8$th odd positive integer after $1$, $3$, $5$, $7$, $9$, $15$, $21$ such that all smaller odd integers greater than $1$ which are coprime to it are prime.
:The $9$th triangular number after $1$, $3$, $6$, $10$, $15$, $21$, $28$, $36$:
::$45 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = \dfrac {9 \times \paren {9 + 1} } 2$
:The $22$nd odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
::$1$, $3$, $5$, $7$, $\ldots$, $35$, $37$, $41$, $43$, $45$, $\ldots$
:The $28$th positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $33$, $37$, $38$, $42$, $43$, $44$ which cannot be expressed as the sum of distinct pentagonal numbers.
:One of the cycle of $5$ numbers to which Kaprekar's process on $2$-digit numbers converges:
::$45 \to 09 \to 81 \to 63 \to 27 \to 45$
== Also see ==
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
== Sources ==
* : $45$
* : $2025$
* : $45$
* : $2025$
Category:Specific Numbers
Category:45
45,946

== Number ==
$45 \, 946$ (forty-five thousand, nine hundred and forty-six) is:
:$2 \times 22 \, 973$
:The $24$th term of a sociable chain of order $28$ whose smallest term is $14 \, 316$:
::$14 \, 316$, $19 \, 116$, $31 \, 704$, $47 \, 616$, $83 \, 328$, $177 \, 792$, $295 \, 488$, $629 \, 072$, $589 \, 786$, $294 \, 896$, $358 \, 336$, $418 \, 904$, $366 \, 556$, $274 \, 924$, $275 \, 444$, $243 \, 760$, $376 \, 736$, $381 \, 028$, $285 \, 778$, $152 \, 990$, $122 \, 410$, $97 \, 946$, $48 \, 976$, $45 \, 946$, $22 \, 976$, $22 \, 744$, $19 \, 916$, $17 \, 716$, $14 \, 316$, $\ldots$
== Also see ==
* 
Category:Specific Numbers
Category:45,946
451

== Number ==
$451$ (four hundred and fifty-one) is:
:$11 \times 41$
:The $6$th Fermat pseudoprime to base $4$ after $15$, $85$, $91$, $341$, $435$:
::$4^{451} \equiv 4 \pmod {451}$
:The smallest positive integer the decimal expansion of whose reciprocal has a period of $10$:
::$\dfrac 1 {451} = 0 \cdotp \dot 00221 \, 7294 \dot 9$
== Historical Note ==
== Also see ==
* 
* 
Category:Specific Numbers
Category:451
4514

== Number ==
$4514$ (four thousand, five hundred and fourteen) is:
:$2 \times 37 \times 61$
:The magic constant of the smallest prime magic square whose elements are consecutive odd primes from $3$ upwards:
== Also see ==
* Magic Constant of Smallest Prime Magic Square with Consecutive Primes from 3
== Sources ==
* : $144$
* : $144$
Category:Specific Numbers
Category:4514
4523

== Number ==
$4523$ (four thousand, five hundred and twenty-three) is:
:The $615$th prime number
:The smallest number which in binary is non-palindromic but with a palindromic square:
::$1000110101011^2 = 1001110000010100000111001$
== Also see ==
* 
== Sources ==
* : $4523$
Category:Specific Numbers
Category:4523
454

== Number ==
$454$ (four hundred and fifty-four) is:
:$2 \times 227$
:The $19$th Smith number after $4$, $22$, $27$, $58$, $85$, $94$, $121$, $166$, $202$, $265$, $274$, $319$, $346$, $355$, $378$, $382$, $391$, $438$:
::$4 + 5 + 4 = 2 + 2 + 2 + 7 = 13$
== Also see ==
* 
Category:Specific Numbers
Category:454
4547

== Number ==
$4547$ (four thousand, five hundred and forty-seven) is:
:The $616$th prime number
:The $12$th prime $p$ such that $p \# + 1$, where $p \#$ denotes primorial (product of all primes up to $p$) of $p$, is prime:
::$2$, $3$, $5$, $7$, $11$, $31$, $379$, $1019$, $1021$, $2657$, $3229$, $4547$
== Also see ==
* 
* 
Category:Specific Numbers
Category:4547
455

== Number ==
$455$ (four hundred and fifty-five) is:
:$5 \times 7 \times 13$
:The $13$th tetrahedral number, after $1$, $4$, $10$, $20$, $35$, $56$, $84$, $120$, $165$, $220$, $286$, $364$:
::$455 = \ds \sum_{k \mathop = 1}^{13} \frac {k \paren {k + 1} } 2 = \dfrac {13 \paren {13 + 1} \paren {13 + 2} } 6$
== Also see ==
* 
Category:Specific Numbers
Category:455
455,052,511

== Number ==
$455 \, 052 \, 511$ is:
:$97 \times 331 \times 14 \, 173$
:The number of primes with no more than $10$ digits
== Also see ==
* 
== Sources ==
* : $455,052,511$
* : $455,052,511$
Category:Specific Numbers
Category:455,052,511
4558

== Number ==
$4558$ (four thousand, five hundred and fifty-eight) is:
:$2 \times 43 \times 53$
:The $43$rd heptagonal number after $1$, $7$, $18$, $34$, $55$, $81$, $112$, $\ldots$, $3010$, $3186$, $3367$, $3744$, $3940$, $4141$, $4347$:
::$4558 = \ds \sum_{k \mathop = 1}^{43} \paren {5 k - 4} = \dfrac {43 \paren {5 \times 43 - 3} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:4558
456

== Number ==
$456$ (four hundred and fifty-six) is:
:$2^3 \times 3 \times 19$
:The $8$th heptagonal pyramidal number after $1$, $8$, $26$, $60$, $115$, $196$, $308$:
::$456 = 1 + 7 + 18 + 34 + 55 + 81 + 112 + 148 = \dfrac {8 \paren {8 + 1} \paren {5 \times 8 - 2} } 6$
== Also see ==
* 
Category:Specific Numbers
Category:456
4560

== Number ==
$4560$ (four thousand, five hundred and fifty-nine) is:
:$2^4 \times 3 \times 5 \times 19$
:The $48$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $3160$, $3321$, $3486$, $3655$, $3828$, $4005$, $4186$, $4371$:
::$4560 = \ds \sum_{k \mathop = 1}^{48} \paren {4 k - 3} = 48 \paren {2 \times 48 - 1}$
:The $95$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $3655$, $3741$, $3828$, $3916$, $4005$, $4095$, $4186$, $4278$, $4371$, $4465$:
::$4560 = \ds \sum_{k \mathop = 1}^{95} k = \dfrac {95 \times \paren {95 + 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:4560
4583

== Number ==
$4583$ (four thousand, five hundred and eighty-three) is:
:The $620$th 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:
::$3$, $5$, $11$, $13$, $41$, $89$, $317$, $337$, $991$, $1873$, $2053$, $2377$, $4093$, $4297$, $4583$
== Also see ==
* 
* 
Category:Specific Numbers
Category:4583
459,818,240

== Number ==
$459 \, 818 \, 240$ (four hundred and fifty-nine million, eight hundred and eighteen thousand, two hundred and forty) is:
:$2^8 \times 5 \times 7 \times 19 \times 37 \times 73$
:The $4$th triperfect number after $120$, $672$, $523 \, 776$:
::$\map {\sigma_1} {459 \, 818 \, 240} = 1 \, 379 \, 454 \, 720 = 3 \times 459 \, 818 \, 240$
== Also see ==
* 
Category:Specific Numbers
Category:459,818,240
46

== Number ==
$46$ (forty-six) is:
:$2 \times 23$
:The $3$rd of the $1$st ordered quadruple of consecutive integers that have divisor sums which are strictly decreasing:
::$\map {\sigma_1} {44} = 84$, $\map {\sigma_1} {45} = 78$, $\map {\sigma_1} {46} = 72$, $\map {\sigma_1} {47} = 48$
:The smallest positive integer which can be expressed as the sum of $2$ distinct lucky numbers in $5$ different ways:
::$46 = 3 + 43 = 9 + 37 = 13 + 33 = 15 + 31 = 21 + 25$
:The $16$th semiprime after $4$, $6$, $9$, $10$, $14$, $15$, $21$, $22$, $25$, $26$, $33$, $34$, $35$, $38$, $39$:
::$46 = 2 \times 23$
:The $29$th positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $33$, $37$, $38$, $42$, $43$, $44$, $45$ which cannot be expressed as the sum of distinct pentagonal numbers.
== Also see ==
* 
* 
* 
* 
== Historical Note ==
Category:Specific Numbers
Category:46
46,233

== Number ==
$46 \, 233$ (forty-six thousand, two hundred and thirty-three) is:
:$3^2 \times 11 \times 467$
:The sum of the first $8$ factorials:
::$46 \, 233 = 1! + 2! + 3! + 4! + 5! + 6! + 7! + 8!$
== Also see ==
* 
Category:Specific Numbers
Category:46,233
46,257,585,588,439

== Number ==
$46 \, 257 \, 585 \, 588 \, 439$ is:
:A prime number
:Has a square root whose continued fraction expansion has the remarkably long period $25 \, 679 \, 652$
== Also see ==
* 
== Sources ==
* 
* : $46,257,585,588,439$
Category:Specific Numbers
Category:46,257,585,588,439
46,656

== Number ==
$46 \, 656$ (forty-six thousand, six hundred and fifty-six) is:
:$2^6 \times 3^6$
:The $6$th power of $6$ after $(1)$, $6$, $36$, $216$, $1296$, $7776$:
::$46 \, 656 = 6^6$
:The $6$th $6$th power after $1$, $64$, $729$, $4096$, $15 \, 625$:
::$46 \, 656 = 6 \times 6 \times 6 \times 6 \times 6 \times 6$
:The $36$th cube number:
::$46 \, 656 = 36 \times 36 \times 36$
:The $216$th square number:
::$46 \, 656 = 216 \times 216$
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:46,656
46,657

== Number ==
$46 \, 657$ (forty-six thousand, six hundred and fifty-seven) is:
:$13 \times 37 \times 97$
:The $6$th Sierpiński number of the first kind after $2$, $5$, $28$, $257$, $3126$:
::$46 \, 657 = 6^6 + 1$
:The $12$th Carmichael number after $561$, $1105$, $1729$, $2465$, $2821$, $6601$, $8911$, $10 \, 585$, $15 \, 841$, $29 \, 341$, $41 \, 041$:
::$\forall a \in \Z: a \perp 46 \, 657: a^{46 \, 657} \equiv 1 \pmod {46 \, 657}$
== Also see ==
* 
* 
Category:Specific Numbers
Category:46,657
46,971

== Number ==
$46 \, 971$ (forty-six thousand, nine hundred and seventy-one) is:
:$3^2 \times 17 \times 307$
:The $6$th triangular number after $28$, $91$, $351$, $2926$, $8001$ which is the sum of $2$ cubes:
::$46 \, 971 = 5096 + 42 \, 875 = 16^3 + 35^3$
:The $306$th triangular number:
::$46 \, 971 = \ds \sum_{k \mathop = 1}^{306} k = \dfrac {306 \times \paren {306 + 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:46,971
46/Historical Note

== Historical Note on 46 ==
Consider the $46$th , as rendered in the of the .
The $46$th word of this psalm is shake.
The $46$th word from the end, counting backwards, is spear.
When the was completed in $1610$, was $46$ years old.
Note also that $1610 = 35 \times 46$.
== Sources ==
* : $46$
* : $46$
Category:Historical Notes
46/Historical Note/Mistake

== Source Work ==
:
:The Dictionary
::$46$
:
:The Dictionary
::$46$
== Mistake ==
:... in Psalm $46$, the $46$th word is 'shake'. The $46$th word from the end counting backwards is 'spear'. Shakespear!
== Correction ==
The correct spelling is in fact Shakespeare.
== Sources ==
* : $46$
* : $46$
Category:Mistakes/Curious and Interesting Numbers
4609

== Number ==
$4609$ (four thousand, six hundred and nine) is:
:$11 \times 419$
:The $9$th Cullen number after $1$, $3$, $9$, $25$, $65$, $161$, $385$, $897$, $2049$:
::$4609 = 9 \times 2^9 + 1$
== Also see ==
* 
Category:Specific Numbers
Category:4609
461

== Number ==
$461$ (four hundred and sixty-one) is:
:The $89$th prime number
:The $5$th integer after $2$, $5$, $17$, $41$ at which the prime number race between primes of the form $4 n + 1$ and $4 n - 1$ are tied.
:The smaller of the $24$th pair of twin primes:
::$463 - 461 = 2$
:The $32$nd long period prime after $7$, $17$, $19$, $23$, $29$, $\ldots$, $269$, $313$, $337$, $367$, $379$, $383$, $389$, $419$, $433$
== Also see ==
* 
* 
* 
* 
== Sources ==
* : $461$
Category:Specific Numbers
Category:461
462

== Number ==
$462$ (four hundred and sixty-two) is:
:$2 \times 3 \times 7 \times 11$
:The index (after $2$, $3$, $6$, $30$, $75$, $81$, $115$, $123$, $249$, $362$, $384$) of the $12$th Woodall prime:
::$462 \times 2^{462} - 1$
:The $2$nd positive integer after $38$ whose square ends in $444$:
::$462^2 = \left({500 - 38}\right)^2 = \left({250 - 38}\right) \times 1000 + 1444 = 213 \, 144$
== Also see ==
* 
* 
== Sources ==
* : $462$
* : $462$
Category:Specific Numbers
Category:462
4624

== Number ==
$4624$ (four thousand, six hundred and twenty-four) is:
:$2^4 \times 17^2$
:The $26$th square number after $1$, $4$, $36$, $121$, $144$, $256$, $\ldots$, $1936$, $2304$, $2704$, $2916$, $3136$, $3600$, $3844$, $4096$, $4356$ to be the divisor sum value of some (strictly) positive integer:
::$4624 = \map {\sigma_1} {4039}$
:The $68$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $3249$, $3364$, $3481$, $3600$, $3721$, $3844$, $3969$, $4096$, $4225$, $4356$, $4489$:
::$4624 = 68 \times 68$
== Also see ==
* 
* 
Category:Specific Numbers
Category:4624
464

== Number ==
$464$ (four hundred and sixty-four) is:
:$2^4 \times 29$
:The $8$th primitive abundant number after $20$, $70$, $88$, $104$, $272$, $304$, $368$:
::$1 + 2 + 4 + 8 + 16 + 29 + 58 + 116 + 232 = 466 > 464$
:The $10$th primitive semiperfect number after $6$, $20$, $28$, $88$, $104$, $272$, $304$, $350$, $368$:
::$464 = 1 + 4 + 8 + 16 + 29 + 58 + 116 + 232$
== Also see ==
* 
* 
Category:Specific Numbers
Category:464
4641

== Number ==
$4641$ (four thousand, six hundred and forty-one) is:
:$3 \times 7 \times 13 \times 17$
:The magic constant of a magic square of order $21$, after $1$, $(5)$, $15$, $34$, $\ldots$, $870$, $1105$, $1379$, $1695$, $2056$, $2465$, $2925$, $3439$, $4010$:
::$\ds 4641 = \dfrac 1 {21} \sum_{k \mathop = 1}^{21^2} k = \dfrac {21 \paren {21^2 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:4641
465

== Number ==
$465$ (four hundred and sixty-five) is:
:$3 \times 5 \times 31$
:The $30$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $325$, $351$, $378$, $406$, $435$:
::$465 = \ds \sum_{k \mathop = 1}^{30} k = \dfrac {30 \times \paren {30 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:465
4656

== Number ==
$4656$ (four thousand, six hundred and fifty-six) is:
:$2^4 \times 3 \times 97$
:The $96$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $4095$, $4186$, $4278$, $4371$, $4465$, $4560$:
::$4656 = \ds \sum_{k \mathop = 1}^{96} k = \dfrac {96 \times \paren {96 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:4656
466

== Number ==
$466$ (four hundred and sixty-six) is:
:$2 \times 233$
:The $44$th noncototient after $10$, $26$, $34$, $50$, $\ldots$, $344$, $346$, $362$, $366$, $372$, $386$, $394$, $404$, $412$, $436$:
::$\nexists m \in \Z_{>0}: m - \phi \left({m}\right) = 466$
:where $\phi \left({m}\right)$ denotes the Euler $\phi$ function
:The largest positive integer which cannot be expressed as the sum of less than $32$ positive fifth powers:
::$466 = 18 \times 1^5 + 14 \times 2^5$
== Also see ==
* Largest Number not Expressible as Sum of Less than 32 Positive Fifth Powers
* 
== Sources ==
* : $466$
Category:Specific Numbers
Category:466
467

== Number ==
$467$ (four hundred and sixty-seven) is:
:The $91$st prime number
:The $16$th safe prime after $5$, $7$, $11$, $23$, $47$, $59$, $83$, $107$, $167$, $179$, $227$, $263$, $347$, $359$, $383$:
::$467 = 2 \times 233 + 1$
:The $33$rd left-truncatable prime after $2$, $3$, $5$, $7$, $\ldots$, $197$, $223$, $283$, $313$, $317$, $337$, $347$, $353$, $367$, $373$, $383$, $397$, $443$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:467
4681

== Number ==
$4681$ (four thousand, six hundred and eighty-one) is:
:$31 \times 151$
:The $16$th Poulet number after $341$, $561$, $645$, $1105$, $1387$, $1729$, $1905$, $2047$, $2465$, $2701$, $2821$, $3277$, $4033$, $4369$, $4371$:
::$2^{4681} \equiv 2 \pmod {4681}$: $4681 = 31 \times 151$
:The $40$th centered hexagonal number after $1$, $7$, $19$, $37$, $61$, $91$, $127$, $\ldots$, $2791$, $2977$, $3169$, $3367$, $3571$, $3781$, $3997$, $4219$, $4447$:
::$4681 = \ds 1 + \sum_{k \mathop = 1}^{40 - 1} 6 k = 40^3 - 39^3$
== Also see ==
* 
* 
Category:Specific Numbers
Category:4681
469

== Number ==
$469$ (four hundred and sixty-nine) is:
:$7 \times 67$
:The $13$th centered hexagonal number after $1$, $7$, $19$, $37$, $61$, $91$, $127$, $169$, $217$, $271$, $331$, $397$:
::$469 = \ds 1 + \sum_{k \mathop = 1}^{13 - 1} 6 k = 13^3 - 12^3$
:The $14$th heptagonal number after $1$, $7$, $18$, $34$, $55$, $81$, $112$, $148$, $189$, $235$, $286$, $342$, $403$:
::$469 = \ds \sum_{k \mathop = 1}^{14} \paren {5 k - 4} = \dfrac {14 \paren {5 \times 14 - 3} } 2$
:The $15$th integer $m$ such that $m! - 1$ (its factorial minus $1$) is prime:
::$3$, $4$, $6$, $7$, $12$, $14$, $30$, $32$, $33$, $38$, $94$, $166$, $324$, $379$, $469$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:469
47

== Number ==
$47$ (forty-seven) is:
:The $15$th prime number, after $2$, $3$, $5$, $7$, $11$, $13$, $17$, $19$, $23$, $29$, $31$, $37$, $41$, $43$
:The $1$st of the largest known pair of Ulam numbers which differ by $1$:
::$47 = 11 + 36, \ 48 = 1 + 47$
:The $1$st number such that the result when adding $2$ is the reversal of when multiplying by $2$:
::$47 + 2 = 49$; $47 \times 2 = 94$
:The $4$th and last of the $1$st ordered quadruple of consecutive integers that have divisor sums which are strictly decreasing:
::$\map {\sigma_1} {44} = 84$, $\map {\sigma_1} {45} = 78$, $\map {\sigma_1} {46} = 72$, $\map {\sigma_1} {47} = 48$
:The $2$nd integer $m$ after $1$ whose cube can be expressed as the sum of $m$ consecutive squares:
::$47^3 = \ds \sum_{k \mathop = 1}^{47} \paren {21 + k}^2$
:The $4$th Keith number after $14$, $19$, $28$:
::$4, 7, 11, 18, 29, 47, \ldots$
:The $4$th Thabit number after $(2)$, $5$, $11$, $23$, and $5$th Thabit prime:
::$47 = 3 \times 2^4 - 1$ 
:The $4$th of $29$ primes of the form $2 x^2 + 29$:
::$2 \times 3^2 + 29 = 47$ ()
:The $5$th safe prime after $5$, $7$, $11$, $23$:
::$47 = 2 \times 23 + 1$
:The $6$th Lucas prime after $2$, $3$, $7$, $11$, $29$
:The $6$th long period prime after $7$, $17$, $19$, $23$, $29$
:The $7$th prime $p$ after $11$, $23$, $29$, $37$, $41$, $43$ such that the Mersenne number $2^p - 1$ is composite
:The $8$th Lucas number after $(2)$, $1$, $3$, $4$, $7$, $11$, $18$, $29$:
::$47 = 18 + 29$
:The $10$th left-truncatable prime after $2$, $3$, $5$, $7$, $13$, $17$, $23$, $37$, $43$
:The $15$th Ulam number after $1$, $2$, $3$, $4$, $6$, $8$, $11$, $13$, $16$, $18$, $26$, $28$, $36$, $38$:
::$47 = 11 + 36$
:The $21$st positive integer which is not the sum of $1$ or more distinct squares:
::$2$, $3$, $6$, $7$, $8$, $11$, $12$, $15$, $18$, $19$, $22$, $23$, $24$, $27$, $28$, $31$, $32$, $33$, $43$, $44$, $47$, $\ldots$
:The $23$rd odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
::$1$, $3$, $5$, $7$, $\ldots$, $35$, $37$, $41$, $43$, $45$, $47$, $\ldots$
== Also see ==
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
== Sources ==
* : $47$
* : $47$
* : $103,823$
Category:Specific Numbers
Category:47
47,616

== Number ==
$47 \, 616$ (forty-seven thousand, six hundred and sixteen) is:
:$2^9 \times 3 \times 31$
:The $4$th term of a sociable chain of order $28$ whose smallest term is $14 \, 316$:
::$14 \, 316$, $19 \, 116$, $31 \, 704$, $47 \, 616$, $83 \, 328$, $177 \, 792$, $295 \, 488$, $629 \, 072$, $589 \, 786$, $294 \, 896$, $358 \, 336$, $418 \, 904$, $366 \, 556$, $274 \, 924$, $275 \, 444$, $243 \, 760$, $376 \, 736$, $381 \, 028$, $285 \, 778$, $152\, 990$, $122 \, 410$, $97 \, 946$, $48 \, 976$, $45 \, 946$, $22 \, 976$, $22 \, 744$, $19 \, 916$, $17 \, 716$, $14 \, 316$, $\ldots$
== Also see ==
* 
Category:Specific Numbers
Category:47,616
47,619

== Number ==
$47 \, 619$ (forty-seven thousand, six hundred and nineteen) is:
:$3^2 \times 11 \times 13 \times 37$
:The recurring part of the Reciprocal of 21:
::$\dfrac 1 {21} = 0 \cdotp \dot 04761 \, \dot 9$
::$047 + 619 = 666$, while $047 \, 619 = 143 \times 333$
::$04 + 76 + 19 = 99$, while $047 \, 619 = 481 \times 99$
::$47 \, 619^2 = 2 \, 267 \, 569 \, 161$, while $2267 + 569 \, 161 = 571 \, 428$, the recurring part of $\dfrac 4 7$
== Also see ==
* Properties of 47,619
== Sources ==
* : $47,619$
* : $47,619$
Category:Specific Numbers
Category:47,619
470

== Number ==
$470$ (four hundred and seventy) is:
:$2 \times 5 \times 47$
:The $45$th noncototient after $10$, $26$, $34$, $50$, $\ldots$, $344$, $346$, $362$, $366$, $372$, $386$, $394$, $404$, $412$, $436$, $466$:
::$\nexists m \in \Z_{>0}: m - \phi \left({m}\right) = 470$
:where $\phi \left({m}\right)$ denotes the Euler $\phi$ function
== Also see ==
* 
Category:Specific Numbers
Category:470
4713

== Number ==
$4713$ (four thousand, seven hundred and thirteen) is:
:$3 \times 1571$
:The index of the $3$rd Cullen prime after $1$, $141$:
::$4713 \times 2^{4713} + 1$
== Also see ==
* 
== Sources ==
* : $4713$
Category:Specific Numbers
Category:4713
472

== Number ==
$472$ (four hundred and seventy-two) is:
:$2^3 \times 59$
:The $5$th positive integer $n$ after $157, 262, 367, 412$ for which $\map \phi {2 n + 1} < \map \phi {2 n}$:
::$\map \phi {2 \times 472 + 1} = 432$, $\map \phi {2 \times 472} = 464$
:The $25$th number whose divisor sum is square:
:
:The $36$th untouchable number after $2$, $5$, $52$, $88$, $96$, $120$, $124$, $\ldots$, $324$, $326$, $336$, $342$, $372$, $406$, $408$, $426$, $430$, $448$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:472
472,335,975

== Number ==
$472 \, 335 \, 975$ is:
:$3^3 \times 5^2 \times 699 \, 757$
:The $29$th pluperfect digital invariant after $1$, $2$, $3$, $4$, $5$, $6$, $\ldots$, $9 \, 926 \, 315$, $24 \, 678 \, 050$, $24 \, 678 \, 051$, $88 \, 593 \, 477$, $146 \, 511 \, 208$:
== Also see ==
* 
Category:Specific Numbers
Category:472,335,975
4720

== Number ==
$4720$ (four thousand, seven hundred and twenty) is:
:$2^4 \times 5 \times 59$
:The $40$th octagonal number, after $1$, $8$, $21$, $40$, $65$, $\ldots$, $3008$, $3201$, $3400$, $3605$, $3816$, $4033$, $4256$, $4485$:
::$4720 = \ds \sum_{k \mathop = 1}^{40} \paren {6 k - 5} = 40 \paren {3 \times 40 - 2}$
== Also see ==
* 
Category:Specific Numbers
Category:4720
4725

== Number ==
$4725$ (four thousand, seven hundred and twenty-five) is:
:$3^3 \times 5^2 \times 7$
:The $7$th odd abundant number after $945$, $1575$, $2205$, $2835$, $3465$, $4095$:
::$\map {\sigma_1} {4725} - 4725 = 5195 > 4725$
== Also see ==
* 
Category:Specific Numbers
Category:4725
474

== Number ==
$474$ (four hundred and seventy-four) is:
:$2 \times 3 \times 79$
:The $37$th untouchable number after $2$, $5$, $52$, $88$, $96$, $120$, $124$, $\ldots$, $324$, $326$, $336$, $342$, $372$, $406$, $408$, $426$, $430$, $448$, $472$
:The $46$th noncototient after $10$, $26$, $34$, $50$, $\ldots$, $344$, $346$, $362$, $366$, $372$, $386$, $394$, $404$, $412$, $436$, $466$, $470$:
::$\nexists m \in \Z_{>0}: m - \map \phi m = 474$
:where $\map \phi m$ denotes the Euler $\phi$ function
== Also see ==
* 
* 
Category:Specific Numbers
Category:474