nyuuzyou/wikis huggingface dataset:
551

== Number ==
$551$ (five hundred and fifty-one) is:
:$19 \times 29$
:The $19$th second pentagonal number after $2$, $7$, $15$, $26$, $40$, $57$, $77$, $100$, $126$, $155$, $187$, $222$, $260$, $301$, $345$, $392$, $442$, $495$:
::$551 = \dfrac {19 \paren {3 \times 19 + 1} } 2$
:The $38$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $392$, $425$, $442$, $477$, $495$, $532$:
::$551 = \dfrac {19 \paren {3 \times 19 + 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:551
552

== Number ==
$552$ (five hundred and fifty-two) is:
:$2^3 \times 3 \times 23$
:The $44$th untouchable number after $2$, $5$, $52$, $88$, $96$, $120$, $124$, $\ldots$, $406$, $408$, $426$, $430$, $448$, $472$, $474$, $498$, $516$, $518$, $520$, $530$, $540$
== Also see ==
* 
Category:Specific Numbers
Category:552
5525

== Number ==
$5525$ (five thousand, five hundred and twenty-five) is:
:$5^2 \times 13 \times 17$
:The $3$rd square pyramorphic number after $1$, $55$:
::$5525 = \ds \sum_{k \mathop = 1}^{25} k^2 = \dfrac {25 \paren {25 + 1} \paren {2 \times 25 + 1} } 6$
:The $25$th square pyramidal number after $1$, $5$, $14$, $30$, $55$, $\ldots$, $1015$, $1240$, $1496$, $1785$, $2109$, $2470$, $2870$, $3311$, $3795$, $4324$, $4900$:
::$5525 = \ds \sum_{k \mathop = 1}^{25} k^2 = \dfrac {25 \paren {25 + 1} \paren {2 \times 25 + 1} } 6$
== Also see ==
* 
* 
Category:Specific Numbers
Category:5525
554,688,278,429

== Number ==
$554 \, 688 \, 278 \, 429$ is:
:A prime number
:A Sophie Germain prime
:The $1$st term of the smallest Cunningham chain of the first kind of length $12$:
::$554 \, 688 \, 278 \, 429$, $1 \, 109 \, 376 \, 556 \, 859$, $2 \, 218 \, 753 \, 113 \, 719$, $4 \, 437 \, 506 \, 227 \, 439$,
:::$8 \, 875 \, 012 \, 454 \, 879$, $17 \, 750 \, 024 \, 909 \, 759$, $35 \, 500 \, 049 \, 819 \, 519$, $71 \, 000 \, 099 \, 639 \, 039$,
:::$142 \, 000 \, 199 \, 278 \, 079$, $284 \, 000 \, 398 \, 556 \, 159$, $568 \, 000 \, 797 \, 112 \, 319$, $1 \, 136 \, 001 \, 594 \, 224 \, 639$
== Also see ==
* 
* 
== Sources ==
* : $554,688,278,429$
Category:Specific Numbers
Category:554,688,278,429
555,555,555,551

== Number ==
$555 \, 555 \, 555 \, 551$ (five hundred and fifty-five billion, five hundred and fifty-five million, five hundred and fifty-five thousand, five hundred and fifty-one) is:
:The $21 \, 366 \, 409 \, 911$th prime number
:The $412$th near-repdigit prime
== Also see ==
* 
* 
Category:Specific Numbers
Category:555,555,555,551
555,555,555,555,556/Mistake

== Source Work ==
:
:The Dictionary
::$555,555,555,555,556$
== Mistake ==
:A Kaprekar number.
== Correction ==
It appears not.
It does not appear possible to divide $555\,555\,555\,555\,556^2$ into two parts which sum to $555\,555\,555\,555\,556$.
This section has been removed from of $1997$.
== Sources ==
* : $555,555,555,555,556$
Category:Mistakes/Curious and Interesting Numbers
556

== Number ==
$556$ (five hundred and fifty-six) is:
:$2^2 \times 139$
:The $45$th untouchable number after $2$, $5$, $52$, $88$, $96$, $120$, $124$, $\ldots$, $406$, $408$, $426$, $430$, $448$, $472$, $474$, $498$, $516$, $518$, $520$, $530$, $540$, $552$
== Also see ==
* 
Category:Specific Numbers
Category:556
5564

== Number ==
$5564$ (five thousand, five hundred and sixty-four) is:
:$2^2 \times 13 \times 107$
:The larger of the $4$th amicable pair, with $5020$:
::$\map {\sigma_1} {5020} = \map {\sigma_1} {5564} = 10 \, 584 = 5020 + 5564$
== Also see ==
* 
== Sources ==
* : $5020$
* : $5020$
Category:Specific Numbers
Category:5564
5565

== Number ==
$5565$ (five thousand, five hundred and sixty-five) is:
:$3 \times 5 \times 7 \times 53$
:The $53$rd hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $4005$, $4186$, $4371$, $4560$, $4950$, $5151$, $5356$:
::$5565 = \ds \sum_{k \mathop = 1}^{53} \paren {4 k - 3} = 53 \paren {2 \times 53 - 1}$
:The $105$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $4560$, $4656$, $4753$, $4851$, $4950$, $5050$, $5151$, $5253$, $5356$, $5460$:
::$5565 = \ds \sum_{k \mathop = 1}^{105} k = \dfrac {105 \times \paren {105 + 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:5565
5566

== Number ==
$5566$ (five thousand, five hundred and fifty-six) is:
:$2 \times 11^2 \times 23$
:The $22$nd pentagonal pyramidal number after $1$, $6$, $12$, $40$, $75$, $\ldots$, $1470$, $1800$, $2176$, $2601$, $3078$, $3610$, $4200$, $4851$:
::$5566 = \ds \sum_{k \mathop = 1}^{22} \dfrac {k \paren {3 k - 1} } 2 = \dfrac {22^2 \paren {22 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:5566
557

== Number ==
$557$ (five hundred and fifty-seven) is:
:The $102$nd prime number
:The larger of the $8$th pair of primes whose prime gap is $10$:
::$557 - 547 = 10$
:The $25$th near-repdigit prime after $101$, $113$, $131$, $151$, $\ldots$, $353$, $373$, $383$, $433$, $443$, $449$, $499$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:557
559

== Number ==
$559$ (five hundred and fifty-nine) is:
:$13 \times 43$
:The $7$th and last positive integer after $79$, $159$, $239$, $319$, $399$, $479$ which cannot be expressed as the sum of fewer than $19$ fourth powers:
::$559 = 15 \times 1^4 + 2 \times 2^4 + 2 \times 4^4$
:or:
::$559 = 9 \times 1^4 + 4 \times 2^4 + 6 \times 3^4$
== Also see ==
* 
== Sources ==
* : $559$
* : $559$
Category:Specific Numbers
Category:559
559 is not Expressible as Sum of Fewer than 19 Fourth Powers

== Theorem ==
$559$ cannot be expressed as the sum of fewer than $19$ fourth powers:
:$559 = 15 \times 1^4 + 2 \times 2^4 + 2 \times 4^4$
or:
:$559 = 9 \times 1^4 + 4 \times 2^4 + 6 \times 3^4$
== Proof ==
First note that $5^4 = 625 > 559$.
Then note that $3 \times 4^4 = 768 > 559$.
Hence any expression of $559$ as fourth powers uses no $n^4$ for $n \ge 5$, and uses not more than $2$ instances 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 sixth use 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:559
56

== Number ==
$56$ (fifty-six) is:
:$2^3 \times 7$
:The $1$st 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 $6$th integer $n$ after $1, 3, 15, 30, 35$ with the property that $\map {\sigma_0} n \divides \map \phi n \divides \map {\sigma_1} n$:
::$\map {\sigma_0} {56} = 8$, $\map \phi {56} = 24$, $\map {\sigma_1} {56} = 120$
:The $6$th tetrahedral number, after $1$, $4$, $10$, $20$, $35$:
::$56 = 1 + 3 + 6 + 10 + 15 + 21 = \dfrac {6 \paren {6 + 1} \paren {6 + 2} } 6$
:The number of integer partitions for $11$:
::$\map p {11} = 56$
:The $13$th semiperfect number after $6$, $12$, $18$, $20$, $24$, $28$, $30$, $36$, $40$, $42$, $48$, $54$:
::$56 = 1 + 2 + 4 + 7 + 14 + 28$
== Also see ==
* 
* 
* 
* 
* 
== Sources ==
* : $56$
* : $56$
Category:Specific Numbers
Category:56
56,598,313

== Number ==
$56 \, 598 \, 313$ (fifty-six million, five hundred and ninety-eight thousand, three hundred and thirteen) is:
:The $3 \, 371 \, 848$th prime number
:The $3$rd prime number $p$ after $3$, $487$ (and largest known) which satisfies the equation $p^2 \divides \paren {10^p - 10}$:
::$56 \, 598 \, 313^2 \divides \paren {10^{56 \, 598 \, 313} - 10}$
== Also see ==
* 
* 
Category:Specific Numbers
Category:56,598,313
560

== Number ==
$560$ (five hundred and sixty) is:
:$2^4 \times 5 \times 7$
:The $14$th octagonal number, after $1$, $8$, $21$, $40$, $65$, $96$, $133$, $176$, $225$, $280$, $341$, $408$, $481$:
::$560 = \ds \sum_{k \mathop = 1}^{14} \paren {6 k - 5} = 14 \paren {3 \times 14 - 2}$
:The $14$th tetrahedral number, after $1$, $4$, $10$, $20$, $35$, $56$, $84$, $120$, $165$, $220$, $286$, $364$, $455$:
::$560 = \ds \sum_{k \mathop = 1}^{14} \frac {k \paren {k + 1} } 2 = \dfrac {14 \paren {14 + 1} \paren {14 + 2} } 6$
== Also see ==
* 
* 
Category:Specific Numbers
Category:560
561

== Number ==
$561$ (five hundred and sixty-one) is:
:$3 \times 11 \times 17$
:The $1$st Carmichael number:
::$\forall a \in \Z: a \perp 561: a^{560} \equiv 1 \pmod {561}$
:The $2$nd Poulet number after $341$:
::$2^{561} \equiv 2 \pmod {561}$: $561 = 3 \times 11 \times 17$
:The $4$th Fermat pseudoprime to base $5$ after $4$, $124$, $217$:
::$5^{561} \equiv 5 \pmod {561}$
:The $7$th Fermat pseudoprime to base $4$ after $15$, $85$, $91$, $341$, $435$, $451$:
::$4^{561} \equiv 4 \pmod {561}$
:The $17$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $120$, $153$, $190$, $231$, $276$, $325$, $378$, $435$, $496$:
::$\ds 561 = \sum_{k \mathop = 1}^{17} \paren {4 k - 3} = 17 \paren {2 \times 17 - 1}$
:The $33$rd triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $325$, $351$, $378$, $406$, $435$, $465$, $496$, $528$:
::$\ds 561 = \sum_{k \mathop = 1}^{33} k = \dfrac {33 \times \paren {33 + 1} } 2$
== Also see ==
* 
* 
* 
* 
* 
* 
== Sources ==
* : $561$
* : $561$
* 
Category:Specific Numbers
Category:561
562

== Number ==
$562$ (five hundred and sixty-two) is:
:$2 \times 281$
:The $24$th Smith number after $4$, $22$, $27$, $58$, $\ldots$, $346$, $355$, $378$, $382$, $391$, $438$, $454$, $483$, $517$, $526$, $535$:
::$5 + 6 + 2 = 2 + 2 + 8 + 1 = 13$
:The $46$th untouchable number after $2$, $5$, $52$, $88$, $96$, $120$, $124$, $\ldots$, $406$, $408$, $426$, $430$, $448$, $472$, $474$, $498$, $516$, $518$, $520$, $530$, $540$, $552$, $556$
== Also see ==
* 
* 
Category:Specific Numbers
Category:562
5625

== Number ==
$5625$ (five thousand, six hundred and twenty-five) is:
:$3^2 \times 5^4$
:The $75$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $4096$, $4225$, $4356$, $4489$, $4624$, $4761$, $4900$, $5041$, $5184$, $5329$, $5476$
::$5625 = 75 \times 75$
== Also see ==
* 
Category:Specific Numbers
Category:5625
563

== Number ==
$563$ (five hundred and sixty-three) is:
:The $103$rd prime number
:The $3$rd and largest known Wilson prime after $5$, $13$:
::$563^2 \divides \paren {563 - 1}! + 1$
:The $9$th balanced prime after $5$, $53$, $157$, $173$, $211$, $257$, $263$, $373$:
::$563 = \dfrac {557 + 569} 2$
:The $19$th safe prime after $5$, $7$, $11$, $23$, $47$, $59$, $83$, $107$, $167$, $179$, $227$, $263$, $347$, $359$, $383$, $467$, $479$, $503$:
::$563 = 2 \times 281 + 1$
== Also see ==
* 
* 
* 
* 
== Sources ==
* : $563$
* : $563$
Category:Specific Numbers
Category:563
567

== Number ==
$567$ (five hundred and sixty-seven) is:
:$3^4 \times 7$
:The $1$st, with $854$, of $2$ positive integers which, together with their square, use all the digits from $1$ to $9$:
::$567^2 = 321 \, 489$
== Also see ==
* 
== Sources ==
* : $567$
* : $567$
Category:Specific Numbers
Category:567
5671

== Number ==
$5671$ (five thousand, six hundred and seventy-one) is:
:$53 \times 107$
:The $106$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $5253$, $5356$, $5460$, $5565$:
::$5671 = \ds \sum_{k \mathop = 1}^{106} k = \dfrac {106 \times \paren {106 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:5671
5677

== Number ==
$5677$ (five thousand, six hundred and seventy-seven) is:
:$7 \times 811$
:The $44$th centered hexagonal number after $1$, $7$, $19$, $37$, $61$, $91$, $127$, $\ldots$, $4219$, $4447$, $4681$, $4921$, $5167$, $5419$:
::$5677 = \ds 1 + \sum_{k \mathop = 1}^{44 - 1} 6 k = 44^3 - 43^3$
== Also see ==
* 
Category:Specific Numbers
Category:5677
5688

== Number ==
$5688$ (five thousand, six hundred and eighty-eight) is:
:$2^2 \times 29 \times 47$
:The $48$th heptagonal number after $1$, $7$, $18$, $34$, $55$, $81$, $112$, $\ldots$, $3744$, $3940$, $4141$, $4347$, $4558$, $4774$, $4995$, $5221$, $5452$:
::$5688 = \ds \sum_{k \mathop = 1}^{48} \paren {5 k - 4} = \dfrac {48 \paren {5 \times 48 - 3} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:5688
57

== Number ==
$57$ (fifty-seven) is:
:$3 \times 19$
: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 nearest vertex (the other two being $65$ and $73$).
:The $6$th second pentagonal number after $2$, $7$, $15$, $26$, $40$:
::$57 = \dfrac {6 \paren {3 \times 6 + 1} } 2$
:The $7$th integer after $7$, $13$, $19$, $35$, $38$, $41$ the decimal representation of whose square can be split into two parts which are each themselves square:
::$57^2 = 3249$; $324 = 18^2$, $9 = 3^2$
:The $12$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $22$, $26$, $35$, $40$, $51$:
::$57 = \dfrac {6 \paren {3 \times 6 + 1} } 2$
:The $18$th Ulam number after $1$, $2$, $3$, $4$, $6$, $8$, $11$, $13$, $16$, $18$, $26$, $28$, $36$, $38$, $47$, $48$, $53$:
::$57 = 4 + 53$
:The $20$th semiprime after $4$, $6$, $9$, $10$, $14$, $15$, $21$, $22$, $25$, $26$, $33$, $34$, $35$, $38$, $39$, $46$, $49$, $51$, $55$:
::$57 = 3 \times 19$
== Also see ==
* Smallest Equilateral Triangle with Internal Point at Integer Distances from Vertices
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:57
57,321

== Number ==
$57 \, 321$ (fifty-seven thousand, three hundred and twenty-one) is:
:$3^3 \times 11 \times 193$
:The smaller of the $2$nd pandigital pair, with $60 \, 984$, of integers each have squares which are themselves pandigital:
::$57 \, 321^2 = 3 \, 285 \, 697 \, 041$, $60 \, 984^2 = 3 \, 719 \, 048 \, 256$
== Also see ==
* 
== Sources ==
* : $57,321$
Category:Specific Numbers
Category:57,321
570

== Number ==
$570$ (five hundred and seventy) is:
:$2 \times 3 \times 5 \times 19$
:The $18$th integer $n$ after $1$, $3$, $15$, $30$, $35$, $56$, $70$, $78$, $105$, $140$, $168$, $190$, $210$, $248$, $264$, $357$, $420$ with the property that $\map {\sigma_0} n \divides \map \phi n \divides \map {\sigma_1} n$:
::$\map {\sigma_0} {570} = 16$, $\map \phi {570} = 144$, $\map {\sigma_1} {570} = 1440$
:The $23$rd integer after $7$, $13$, $19$, $35$, $38$, $\ldots$, $223$, $253$, $285$, $305$, $350$, $380$, $410$, $475$, $487$ the decimal representation of whose square can be split into two parts which are each themselves square:
::$570^2 = 324 \, 900$; $324 = 18^2$, $900 = 30^2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:570
571

== Number ==
$571$ (five hundred and seventy-one) is:
:The $105$th prime number
:The larger of the $26$th pair of twin primes:
::$571 - 569 = 2$
:The $39$th long period prime after $7$, $17$, $19$, $23$, $29$, $\ldots$, $383$, $389$, $419$, $433$, $461$, $487$, $491$, $499$, $503$, $509$, $541$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:571
572

== Number ==
$572$ (five hundred and seventy-two) is:
:$2^2 \times 11 \times 13$
:The $10$th primitive abundant number after $20$, $70$, $88$, $104$, $272$, $304$, $368$, $464$, $550$:
::$1 + 2 + 4 + 11 + 13 + 22 + 26 + 44 + 52 + 143 + 286 = 604 > 572$
:The $14$th primitive semiperfect number after $6$, $20$, $28$, $88$, $104$, $272$, $304$, $350$, $368$, $464$, $490$, $496$, $550$:
::$572 = 1 + 11 + 13 + 22 + 44 + 52 + 143 + 286$
== Also see ==
* 
* 
Category:Specific Numbers
Category:572
5720

== Number ==
$5720$ (five thousand, seven hundred and twenty) is:
:$2^3 \times 5 \times 11 \times 13$
:The $44$th octagonal number, after $1$, $8$, $21$, $40$, $65$, $\ldots$, $3816$, $4033$, $4256$, $4485$, $4720$, $4961$, $5208$, $5461$:
::$5720 = \ds \sum_{k \mathop = 1}^{44} \paren {6 k - 5} = 44 \paren {3 \times 44 - 2}$
== Also see ==
* 
Category:Specific Numbers
Category:5720
573

== Number ==
$573$ (five hundred and seventy-three) is:
:$3 \times 191$
:The $3$rd integer after $91$, $428$ whose square has a decimal representation consisting of the concatenation of $2$ consecutive integers:
::$573^2 = 328 \, 329$
:The $2$nd integer after $428$ whose square has a decimal representation consisting of the concatenation of $2$ consecutive increasing integers:
::$573^2 = 328 \, 329$
== Also see ==
* 
* 
Category:Specific Numbers
Category:573
576

== Number ==
$576$ (five hundred and seventy-six) is:
:$2^6 \times 3^2$
:The $9$th square number after $1$, $4$, $36$, $121$, $144$, $256$, $324$, $400$ to be the divisor sum value of some (strictly) positive integer:
::$576 = \map {\sigma_1} {210} = \map {\sigma_1} {282} = \map {\sigma_1} {310} = \map {\sigma_1} {345} = \map {\sigma_1} {357} = \map {\sigma_1} {382} = \map {\sigma_1} {385}$
:The $11$th positive integer after $128$, $192$, $256$, $288$, $320$, $384$, $432$, $448$, $480$, $512$ with $7$ or more prime factors:
::$512 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \paren {\times \, 3}$
:The $24$th square number after $1$, $4$, $8$, $9$, $16$, $25$, $\ldots$, $361$, $392$, $400$, $432$, $441$, $484$, $529$:
::$576 = 24 \times 24$
:The $25$th Smith number after $4$, $22$, $27$, $58$, $\ldots$, $346$, $355$, $378$, $382$, $391$, $438$, $454$, $483$, $517$, $526$, $535$, $562$:
::$5 + 7 + 6 = 2 + 2 + 2 + 2 + 2 + 2 + 3 + 3 = 18$
:The $41$st powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $\ldots$, $400$, $432$, $441$, $484$, $500$, $512$, $529$:
::$576 = 2^6 \times 3^2$
:The $47$th untouchable number after $2$, $5$, $52$, $88$, $96$, $120$, $124$, $\ldots$, $472$, $474$, $498$, $516$, $518$, $520$, $530$, $540$, $552$, $556$, $562$
== Also see ==
* 
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:576
577

== Number ==
$577$ (five hundred and seventy-seven) is:
:The $106$th prime number
:The $7$th positive integer $n$ after $157$, $262$, $367$, $412$, $472$, $487$ for which $\map \phi {2 n + 1} < \map \phi {2 n}$:
::$\map \phi {2 \times 577 + 1} = 480$, $\map \phi {2 \times 577} = 576$
:The smaller of the $9$th pair of primes whose prime gap is $10$:
::$587 - 577 = 10$
:The $9$th prime number of the form $n^2 + 1$ after $2$, $5$, $17$, $37$, $101$, $197$, $257$, $401$:
::$577 = 24^2 + 1$
:The $13$th Proth prime after $3$, $5$, $13$, $17$, $41$, $97$, $113$, $193$, $241$, $257$, $353$, $449$:
::$577 = 9 \times 2^6 + 1$
:The $26$th near-repdigit prime after $101$, $113$, $131$, $151$, $\ldots$, $353$, $373$, $383$, $433$, $443$, $449$, $499$, $557$
:The $40$th long period prime after $7$, $17$, $19$, $23$, $29$, $\ldots$, $383$, $389$, $419$, $433$, $461$, $487$, $491$, $499$, $503$, $509$, $541$, $571$
== Also see ==
* 
* 
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:577
5775

== Number ==
$5775$ (five thousand, seven hundred and seventy-five) is:
:$3 \times 5^2 \times 7 \times 11$
:The $9$th odd abundant number after $945$, $1575$, $2205$, $2835$, $3465$, $4095$, $4725$, $5355$:
::$\map {\sigma_1} {5775} - 5775 = 6129 > 5775$
== Also see ==
* 
Category:Specific Numbers
Category:5775
5776

== Number ==
$5776$ (five thousand, seven hundred and seventy-six) is:
:$2^4 \times 19^2$
:The $29$th square number after $1$, $4$, $36$, $121$, $144$, $256$, $\ldots$, $1936$, $2304$, $2704$, $2916$, $3136$, $3600$, $3844$, $4096$, $4356$, $4624$, $4900$, $5184$ to be the divisor sum value of some (strictly) positive integer:
::$5776 = \map {\sigma_1} {5587}$
:The $76$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $5041$, $5184$, $5329$, $5476$, $5625$
::$5776 = 76 \times 76$
== Also see ==
* 
* 
Category:Specific Numbers
Category:5776
5777

== Number ==
$5777$ (five thousand, seven hundred and seventy-seven) is:
:$53 \times 109$
:The $9$th Stern number after $1$, $3$, $17$, $137$, $227$, $977$, $1187$, $1493$
:The smaller of the only two known composite Stern numbers, with $5993$
== Also see ==
* 
== Sources ==
* : $5777$
* : $5777$
Category:Specific Numbers
Category:5777
5778

== Number ==
$5778$ (five thousand, seven hundred and seventy-eight) is:
:$2 \times 3^3 \times 107$
:The $3$rd and last Lucas number after $1$, $3$ which is also triangular:
::$5778 = \ds \sum_{k \mathop = 1}^{107} k = \dfrac {107 \times \paren {107 + 1} } 2 = 2207 + 3571$
:The $18$th Lucas number after $(2)$, $1$, $3$, $4$, $7$, $11$, $18$, $29$, $47$, $76$, $123$, $199$, $322$, $521$, $843$, $1364$, $2207$, $3571$:
::$5778 = 2207 + 3571$
:The $54$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $4005$, $4186$, $4371$, $4560$, $4950$, $5151$, $5356$, $5565$:
::$5778 = \ds \sum_{k \mathop = 1}^{54} \paren {4 k - 3} = 54 \paren {2 \times 54 - 1}$
:The $107$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $5253$, $5356$, $5460$, $5565$, $5671$:
::$5778 = \ds \sum_{k \mathop = 1}^{107} k = \dfrac {107 \times \paren {107 + 1} } 2$
== Also see ==
* 
* 
* 
* 
== Sources ==
* : $5778$
Category:Specific Numbers
Category:5778
5795

== Number ==
$5795$ (five thousand, seven hundred and ninety-five) is:
:$5 \times 19 \times 61$
:The index of the $4$th Cullen prime after $1$, $141$, $4713$:
::$5795 \times 2^{5795} + 1$
== Also see ==
* 
Category:Specific Numbers
Category:5795
58

== Number ==
$58$ (fifty-eight) is:
:$2 \times 29$
:The $3$rd non-square positive integer which cannot be expressed as the sum of a square and a prime:
::$10$, $34$, $58$, $\ldots$
:The $4$th Smith number after $4$, $22$, $27$:
::$5 + 8 = 2 + 2 + 9 = 13$
:The $5$th positive integer which cannot be expressed as the sum of a square and a prime:
::$1$, $10$, $25$, $34$, $58$, $\ldots$
:The $6$th noncototient after $10$, $26$, $34$, $50$, $52$:
::$\nexists m \in \Z_{>0}: m - \map \phi m = 58$
:where $\map \phi m$ denotes the Euler $\phi$ function
:The $16$th (and probably largest) integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $9$, $10$, $11$, $17$, $18$, $30$, $33$ such that $5^n$ contains no zero in its decimal representation:
::$5^{58} = 34 \, 694 \, 469 \, 519 \, 536 \, 141 \, 888 \, 238 \, 489 \, 627 \, 838 \, 134 \, 765 \, 625$
:The $21$st semiprime after $4$, $6$, $9$, $10$, $14$, $15$, $21$, $22$, $25$, $26$, $33$, $34$, $35$, $38$, $39$, $46$, $49$, $51$, $55$, $58$:
::$58 = 2 \times 29$
== Also see ==
* 
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:58
58,413

== Number ==
$58 \, 413$ (fifty-eight thousand, four hundred and thirteen) is:
:$3 \times 19 \, 471$
:The smaller of the $3$rd pandigital pair, with $96 \, 702$, of integers each have squares which are themselves pandigital:
::$58 \, 413^2 = 3 \, 412 \, 078 \, 569$, $96 \, 702^2 = 9 \, 351 \, 276 \, 804$
== Also see ==
* 
== Sources ==
* : $57,321$
Category:Specific Numbers
Category:58,413
58,786

== Number ==
$58 \, 786$ (fifty-eight thousand, seven hundred and eighty-six) is:
:$2 \times 7 \times 13 \times 17 \times 19$
:The $11$th Catalan number after $(1)$, $1$, $2$, $5$, $14$, $42$, $132$, $429$, $1430$, $4862$, $16 \, 796$:
::$58 \, 786 = \dfrac 1 {11 + 1} \dbinom {2 \times 11} {11} = \dfrac 1 {12} \times 705 \, 432$
== Also see ==
* 
Category:Specific Numbers
Category:58,786
5823

== Number ==
$5823$ (five thousand, eight hundred and twenty-three) is:
:$3^2 \times 647$
: The $5$th integer after $954$, $1980$, $2961$, $3870$ whose digits are unchanged when subtracting its reversal:
::$5823 - 3285 = 2538$
== Also see ==
* 
Category:Specific Numbers
Category:5823
583

== Number ==
$583$ (five hundred and eighty-three) is:
:$11 \times 53$
:The $4$th of the $3$-digit integers $m$ which need the largest number of reverse-and-add process iterations ($23$) before reaching a palindromic number:
::$583, 968, 1837, \ldots, 8713200023178$
== Also see ==
* 
Category:Specific Numbers
Category:583
5830

== Number ==
$5830$ (five thousand, eight hundred and thirty) is:
:$2 \times 5 \times 11 \times 53$
:The $4$th weird number after $70$, $836$, $4030$:
::$\map {\sigma_1} {5830} - 5830 = 5834$: $5830$ cannot be made from its aliquot parts
== Also see ==
* 
Category:Specific Numbers
Category:5830
5832

== Number ==
$5832$ (five thousand, eight hundred and thirty-two) is:
:$2^3 \times 3^6$
:The $4$th cube after $6^3$, $9^3$, $12^3$ which can be expressed as the sum of $3$ positive cubes
:The $5$th cube after $0$, $1$, $512$, $4913$ whose digits add up to its root:
::$5832 = 18^3$, while $5 + 8 + 3 + 2 = 18$
:The $18$th cube number after $1$, $8$, $27$, $64$, $125$, $216$, $343$, $512$, $729$, $1000$, $1331$, $1728$, $2197$, $2744$, $3375$, $4096$, $4913$:
::$5832 = 18 \times 18 \times 18$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:5832