nyuuzyou/wikis huggingface dataset:
246,642

== Number ==
$246 \, 642$ (two hundred and forty-six thousand, six hundred and forty-two) is:
:$2 \times 3 \times 11 \times 37 \times 101$
:The $11$th palindromic integer after $0$, $1$, $2$, $3$, $11$, $77$, $363$, $1111$, $2662$, $111 \, 111$ which is the index of a palindromic triangular number
::$T_{246 \, 642} = 30 \, 416 \, 261 \, 403$
== Also see ==
* 
Category:Specific Numbers
Category:246,642
2465

== Number ==
$2465$ (two thousand, four hundred and sixty-five) is:
:$5 \times 17 \times 29$
:The $4$th Carmichael number after $561$, $1105$, $1729$:
::$\forall a \in \Z: a \perp 2465: a^{2464} \equiv 1 \pmod {2465}$
:The $9$th Poulet number after $341$, $561$, $645$, $1105$, $1387$, $1729$, $1905$, $2047$:
::$2^{2465} \equiv 2 \pmod {2465}$: $2465 = 5 \times 17 \times 29$
:The $11$th Fermat pseudoprime to base $3$ after $91$, $121$, $286$, $671$, $703$, $949$, $1105$, $1541$, $1729$, $1891$:
::$3^{2465} \equiv 3 \pmod {2465}$
:The magic constant of a magic square of order $17$, after $1$, $(5)$, $15$, $34$, $\ldots$, $870$, $1105$, $1379$, $1695$, $2056$:
::$2465 = \ds \dfrac 1 {17} \sum_{k \mathop = 1}^{17^2} k = \dfrac {17 \paren {17^2 + 1} } 2$
:The $29$th octagonal number, after $1$, $8$, $21$, $40$, $65$, $\ldots$, $1281$, $1408$, $1541$, $1680$, $1825$, $1976$, $2133$, $2296$:
::$2465 = \ds \sum_{k \mathop = 1}^{29} \paren {6 k - 5} = 29 \paren {3 \times 29 - 2}$
== Also see ==
* 
* 
* 
* 
* 
== Sources ==
* : $2465$
Category:Specific Numbers
Category:2465
247

== Number ==
$247$ (two hundred and forty-seven) is:
:$13 \times 19$
:The $13$th pentagonal number after $1$, $5$, $12$, $22$, $35$, $51$, $70$, $92$, $117$, $145$, $176$, $210$:
::$247 = 1 + 4 + 7 + 10 + 13 + 16 + 19 + 22 + 25 + 28 + 31 + 34 + 37 = \dfrac {13 \paren {3 \times 13 - 1} } 2$
:The $25$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $100$, $117$, $126$, $145$, $155$, $176$, $187$, $210$, $222$:
::$247 = \dfrac {13 \paren {3 \times 13 - 1} } 2$
:The $48$th positive integer $n$ such that no factorial of an integer can end with $n$ zeroes.
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:247
2470

== Number ==
$2470$ (two thousand, four hundred and seventy) is:
:$2 \times 5 \times 13 \times 19$
:The $19$th square pyramidal number after $1$, $5$, $14$, $30$, $55$, $\ldots$, $1015$, $1240$, $1496$, $1785$, $2109$:
::$2470 = \ds \sum_{k \mathop = 1}^{19} k^2 = \dfrac {19 \paren {19 + 1} \paren {2 \times 19 + 1} } 6$
== Also see ==
* 
Category:Specific Numbers
Category:2470
248

== Number ==
$248$ (two hundred and forty-eight) is:
:$2^3 \times 31$
:The $5$th integer after $1$, $14$, $30$, $105$ whose divisor sum divided by its Euler $\phi$ value is a square:
::$\dfrac {\map {\sigma_1} {248} } {\map \phi {248} } = \dfrac {480} {120} = 4 = 2^2$
:The $14$th integer $n$ after $1, 3, 15, 30, 35, 56, 70, 78, 105, 140, 168, 190, 210$ with the property that $\map {\sigma_0} n \divides \map \phi n \divides \map {\sigma_1} n$:
::$\map {\sigma_0} {248} = 8$, $\map \phi {248} = 120$, $\map {\sigma_1} {248} = 480$
:The $16$th untouchable number after $2$, $5$, $52$, $88$, $96$, $120$, $124$, $146$, $162$, $188$, $206$, $210$, $216$, $238$, $246$
:The $39$th nontotient:
::$\nexists m \in \Z_{>0}: \map \phi m = 248$
:where $\map \phi m$ denotes the Euler $\phi$ function
:The $49$th positive integer $n$ such that no factorial of an integer can end with $n$ zeroes.
== Also see ==
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:248
248,832

== Number ==
$248 \, 832$ (two hundred and forty-eight thousand, eight hundred and thirty-two) is:
:$2^{10} \times 3^5$
:The $1$st fifth power which is the sum of $6$ fifth powers:
::$248 \, 832 = 12^5 = 4^5 + 5^5 + 6^5 + 7^5 + 9^5 + 11^5$
:The $5$th power of $12$ after $(1)$, $12$, $144$, $1728$, $20 \, 736$:
::$248 \, 832 = 12^5$
:The $12$th fifth power after $1$, $32$, $243$, $1024$, $3125$, $7776$, $16 \, 807$, $32 \, 768$, $59 \, 049$, $100 \, 000$, $161 \, 051$:
::$248 \, 832 = 12 \times 12 \times 12 \times 12 \times 12$
== Also see ==
* Smallest Fifth Power which is Sum of 6 Fifth Powers
* 
* 
== Sources ==
* : $248,832$
* : $248,832$
Category:Specific Numbers
Category:248,832
2485

== Number ==
$2485$ (two thousand, four hundred and eighty-five) is:
:$5 \times 7 \times 71$
:The $70$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $1891$, $1953$, $2016$, $2080$, $2145$, $2211$, $2278$, $2346$, $2415$:
::$2485 = \ds \sum_{k \mathop = 1}^{70} k = \dfrac {70 \times \paren {70 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:2485
2487

== Number ==
$2487$ (two thousand, four hundred and eighty-seven) is:
:$3 \times 829$
:The $3$rd positive integer after $353$, $651$ such that its fourth power is the sum of $4$ other fourth powers of positive integers with no common factors:
::$2487^4 = 435^4 + 710^4 + 1384^4 + 2420^4$
== Also see ==
* 
Category:Specific Numbers
Category:2487
249

== Number ==
$249$ (two hundred and forty-nine) is:
:$3 \times 83$
:The index (after $2$, $3$, $6$, $30$, $75$, $81$, $115$, $123$) of the $9$th Woodall prime:
::$249 \times 2^{249} - 1$
:The $14$th trimorphic number after $1$, $4$, $5$, $6$, $9$, $24$, $25$, $49$, $51$, $75$, $76$, $99$, $125$:
::$249^3 = 15 \, 438 \, \mathbf {249}$
== Also see ==
* 
* 
Category:Specific Numbers
Category:249
24 divides Square of Odd Integer Not Divisible by 3 plus 23

== Theorem ==
Let $a \in \Z$ be an integer such that:
:$2 \nmid a$
:$3 \nmid a$
where $\nmid$ denotes non-divisibility.
Then:
:$24 \divides \paren {a^2 + 23}$
where $\divides$ denotes divisibility.
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : Chapter $2$: Divisibility Theory in the Integers: $2.2$ The Greatest Common Divisor: Problems $2.2$: $15 \ \text {(c)}$
Category:24 divides Square of Odd Integer Not Divisible by 3 plus 23
Category:Odd Squares
24 divides Square of Odd Integer Not Divisible by 3 plus 23/Proof 2

== Theorem ==
== Proof ==
Let $a$ be as defined.
We have that $a$ is of the form:
:$a = 6 k + 1$
or:
:$a = 6 k + 5$
Hence:
or:
Category:24 divides Square of Odd Integer Not Divisible by 3 plus 23
24 divides a(a^2 - 1) when a is Odd

== Theorem ==
Let $a \in \Z$ be an odd integer.
Then:
:$24 \divides a \paren {a^2 - 1}$
where $\divides$ denotes divisibility.
== Proof ==
First suppose that $a$ is not divisible by $3$.
Then from Square Modulo 24 of Odd Integer Not Divisible by 3:
:$24 \divides \paren {a^2 - 1}$
from which the result follows immediately.
Now suppose that $3 \divides a$.
Then immediately:
:$3 \divides a \paren {a^2 - 1}$
From Odd Square Modulo 8:
:$8 \divides \paren {a^2 - 1}$
and so:
:$8 \divides a \paren {a^2 - 1}$
We have from Coprime Integers: $3$ and $8$ that:
:$3 \perp 8$
where $\perp$ denotes coprimality.
As we have that:
:$8 \divides a \paren {a^2 - 1}$
and:
:$3 \divides a \paren {a^2 - 1}$
it follows from Product of Coprime Factors that:
:$24 \divides a \paren {a^2 - 1}$
== Sources ==
* : Chapter $2$: Divisibility Theory in the Integers: $2.2$ The Greatest Common Divisor: Problems $2.2$: $15 \ \text {(a)}$
Category:Examples of Divisors of Integers
24 is Smallest Composite Number the Product of whose Proper Divisors is Cube/Mistake

== Source Work ==
:
:The Dictionary
::$24$
:
:The Dictionary
::$24$
== Mistake ==
:The smallest composite number, the product of whose proper divisors is a cube. $2 \times 3 \times 4 \times 6 \times 8 \times 12 = 24^3$.
== Correction ==
There are smaller such numbers.
For example, the product of the proper divisors of $8$ is $8$ itself, which is a cube:
:$2 \times 4 = 8 = 2^3$
What it should say is:
:The smallest composite number, the product of whose proper divisors is its cube. $2 \times 3 \times 4 \times 6 \times 8 \times 12 = 24^3$.
== Sources ==
* : $24$
* : $24$
Category:Mistakes/Curious and Interesting Numbers
25,200

== Number ==
$25 \, 200$ (twenty-five thousand, two hundred) is:
:$2^4 \times 3^2 \times 5^2 \times 7$
:The smallest integer which can be expressed as the product of $4$ different triplets of integers each of which has the same sum:
::$25 \, 200 = 6 \times 56 \times 75 = 7 \times 40 \times 90 = 9 \times 28 \times 100 = 12 \times 20 \times 105$
== Also see ==
* Smallest Integer which is Product of 4 Triples all with Same Sum
== Sources ==
* : $25,200$
Category:Specific Numbers
Category:25,200
25,930

== Number ==
$25 \, 930$ (twenty-five thousand, nine hundred and thirty) is:
:$2 \times 5 \times 2593$
:The $1$st of $4$ integers whose Euler $\phi$ value is $2^7 \times 3^4 = 10 \, 368$
::$\map \phi {25 \, 930} = \map \phi {25 \, 935} = \map \phi {25 \, 940} = \map \phi {25 \, 942} = 2^7 \times 3^4$
== Also see ==
* 
== Sources ==
* : $25,930$
Category:Specific Numbers
Category:25,930
25,935

== Number ==
$25 \, 935$ (twenty-five thousand, nine hundred and thirty-five) is:
:$3 \times 5 \times 7 \times 13 \times 19$
:The $2$nd of $4$ integers whose Euler $\phi$ value is $2^7 \times 3^4 = 10 \, 368$
::$\map \phi {25 \, 930} = \map \phi {25 \, 935} = \map \phi {25 \, 940} = \map \phi {25 \, 942} = 2^7 \times 3^4$
== Also see ==
* 
== Sources ==
* : $25,930$
Category:Specific Numbers
Category:25,935
25,940

== Number ==
$25 \, 940$ (twenty-five thousand, nine hundred and forty) is:
:$2^2 \times 5 \times 1297$
:The $3$rd of $4$ integers whose Euler $\phi$ value is $2^7 \times 3^4 = 10 \, 368$
::$\phi \left({25 \, 930}\right) = \phi \left({25 \, 935}\right) = \phi \left({25 \, 940}\right) = \phi \left({25 \, 942}\right) = 2^7 \times 3^4$
== Also see ==
* 
== Sources ==
* : $25,930$
Category:Specific Numbers
Category:25,940
25,942

== Number ==
$25 \, 942$ (twenty-five thousand, nine hundred and forty-two) is:
:$2 \times 7 \times 17 \times 109$
:The $4$th of $4$ integers whose Euler $\phi$ value is $2^7 \times 3^4 = 10 \, 368$
::$\map \phi {25 \, 930} = \map \phi {25 \, 935} = \map \phi {25 \, 940} = \map \phi {25 \, 942} = 2^7 \times 3^4$
== Also see ==
* 
== Sources ==
* : $25,930$
Category:Specific Numbers
Category:25,942
250

== Number ==
$250$ (two hundred and fifty) is:
:$2 \times 5^3$
:The $12$th positive integer after $50$, $65$, $85$, $125$, $130$, $145$, $170$, $185$, $200$, $205$, $221$ which can be expressed as the sum of two square numbers in two or more different ways:
::$250 = 15^2 + 5^2 = 13^2 + 9^2$
:and the $2$nd such which is also the sum of $2$ cube numbers:
::$250 = 15^2 + 5^2 = 13^2 + 9^2 = 5^3 + 5^3$
== Also see ==
* 
* 
== Sources ==
* : $250$
Category:Specific Numbers
Category:250
2500

== Number ==
$2500$ (two thousand, five hundred) is:
:$2^2 \times 5^4$
:The $50$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $1521$, $1600$, $1681$, $1764$, $1849$, $1936$, $2025$, $2116$, $2209$, $2304$, $2401$:
::$2500 = 50 \times 50$
:The $43$rd positive integer which cannot be expressed as the sum of a square and a prime:
::$1$, $10$, $25$, $34$, $58$, $64$, $85$, $\ldots$, $1024$, $1089$, $1225$, $1255$, $1351$, $1414$, $1444$, $1521$, $1681$, $1849$, $1906$, $1936$, $2116$, $2209$, $2304$, $2500$, $\ldots$
== Also see ==
* 
* 
Category:Specific Numbers
Category:2500
2501

== Number ==
$2501$ (two thousand, five hundred and one) is:
:$41 \times 61$
:The $4$th positive integer after $353$, $651$, $2487$ such that its fourth power is the sum of $4$ other fourth powers of positive integers with no common factors:
::$2501^4 = 1130^4 + 1190^4 + 1432^4 + 2365^4$
:The $41$st pentagonal number after $1$, $5$, $12$, $22$, $35$, $\ldots$, $1820$, $1926$, $2035$, $2147$, $2262$, $2380$:
::$2501 = \ds \sum_{k \mathop = 1}^{41} \paren {3 k - 2} = \dfrac {41 \paren {3 \times 41 - 1} } 2$
:The $81$st generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $2072$, $2147$, $2185$, $2262$, $2301$, $2380$, $2420$:
::$2501 = \ds \sum_{k \mathop = 1}^{41} \paren {3 k - 2} = \dfrac {41 \paren {3 \times 41 - 1} } 2$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:2501
251

== Number ==
$251$ (two hundred and fifty-one) is:
:The $54$th prime number.
:The $1$st positive integer which can be expressed as the sum of $3$ cubes numbers in two or more different ways:
::$251 = 5^3 + 5^3 + 1^3 = 6^3 + 3^3 + 2^3$
:The larger of the $3$rd pair of primes whose prime gap is $10$:
::$251 - 241 = 10$
:The $4$th obstinate number after $1$, $127$, $149$
:The $15$th trimorphic number after $1$, $4$, $5$, $6$, $9$, $24$, $25$, $49$, $51$, $75$, $76$, $99$, $125$, $249$:
::$251^3 = 15 \, 813 \, \mathbf {251}$
:The $18$th Sophie Germain prime after $2$, $3$, $5$, $11$, $23$, $29$, $41$, $53$, $83$, $89$, $113$, $131$, $173$, $179$, $191$, $233$, $239$:
::$2 \times 251 + 1 = 503$, which is prime.
== Also see ==
* 
* 
* 
* 
* 
== Sources ==
* : $251$
* : $251$
Category:Specific Numbers
Category:251
251,133,297

== Number ==
$251 \, 133 \, 297$ is:
:$3 \times 23 \times 1051 \times 3463$
:The $41$st Keith number:
::$2$, $5$, $1$, $1$, $3$, $3$, $2$, $9$, $7$, $33$, $\ldots$, $62 \, 907 \, 224$, $125 \, 690 \, 487$, $251 \, 133 \, 297$, $\ldots$
== Also see ==
* 
== Sources ==
* : $129,572,008$
Category:Specific Numbers
Category:251,133,297
2512

== Number ==
$2512$ (two thousand, five hundred and twelve) is:
:$2^4 \times 157$
:The $32$nd heptagonal number after $1$, $7$, $18$, $34$, $55$, $81$, $112$, $\ldots$, $1525$, $1651$, $1782$, $1918$, $2059$, $2205$, $2356$:
::$2512 = \ds \sum_{k \mathop = 1}^{32} \paren {5 k - 4} = \dfrac {32 \paren {5 \times 32 - 3} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:2512
252

== Number ==
$252$ (two hundred and fifty-two) is:
:$2^2 \times 3^2 \times 7$
:The $5$th central binomial coefficient after $2$, $6$, $20$, $70$:
::$252 = \dbinom {2 \times 5} 5 := \dfrac {10!} {\paren {5!}^2}$
:The $7$th hexagonal pyramidal number after $1$, $7$, $22$, $50$, $95$, $161$:
::$252 = \ds \sum_{k \mathop = 1}^7 k \paren {2 k - 1} = \dfrac {7 \paren {7 + 1} \paren {4 \times 7 - 1} } 6$
:The $16$th of the $17$ positive integers for which the value of the Euler $\phi$ function is $72$:
::$73$, $91$, $95$, $111$, $117$, $135$, $146$, $148$, $152$, $182$, $190$, $216$, $222$, $228$, $234$, $252$, $270$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:252
2520

== Number ==
$2520$ (two thousand, five hundred and twenty) is:
:$2^3 \times 3^2 \times 5 \times 7$
:and so is the lowest common multiple of all the digits from $1$ to $9$
:The smallest EPORN:
::$2520 = 210 \times 012 = 120 \times 021$
:The $6$th and last special highly composite number after $1$, $2$, $6$, $12$, $60$
:The $18$th highly composite number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$, $60$, $120$, $180$, $240$, $360$, $720$, $840$, $1260$, $1680$:
::$\map {\sigma_0} {2520} = 48$
:The $18$th superabundant number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$, $60$, $120$, $180$, $240$, $360$, $720$, $840$, $1260$, $1680$:
::$\dfrac {\map {\sigma_1} {2520} } {2520} = \dfrac {9360} {2520} \approx 3 \cdotp 714$
:The sum of $4$ of its divisors in $6$ different ways:
== Also see ==
* EPORN/Examples/2520
* 2520 equals Sum of 4 Divisors in 6 Ways
* 
* 
* 
== Sources ==
* : $2520$
* : The Ladies' Diary or Woman's Almanac, : $140$
* : $2520$
Category:Specific Numbers
Category:2520
2520 equals Sum of 4 Divisors in 6 Ways

== Theorem ==
The number $2520$ can be expressed as the sum of $4$ of its divisors in $6$ different ways:
This is the maximum possible number of ways it is possible to express an integer as the sum of $4$ of its divisors.
== Proof ==
We apply 1 can be Expressed as Sum of 4 Distinct Unit Fractions in 6 Ways:
Find the maximum powers of the primes in each equation, and choose the largest that appears:
Therefore the smallest number would be:
:$2^3 \times 3^2 \times 5^1 \times 7^1 = 2520$
== Sources ==
* : $2520$
* : $2520$
Category:Recreational Mathematics
Category:2520
253

== Number ==
$253$ (two hundred and fifty-three) is:
:$11 \times 23$
:The $15$th integer after $7$, $13$, $19$, $35$, $38$, $41$, $57$, $65$, $70$, $125$, $130$, $190$, $205$, $223$ the decimal representation of whose square can be split into two parts which are each themselves square:
::$253^2 = 64 \, 009$; $6400 = 80^2$, $9 = 3^2$
:The $22$nd triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $136$, $153$, $171$, $190$, $210$, $231$:
::$253 = \ds \sum_{k \mathop = 1}^{22} k = \dfrac {22 \times \paren {22 + 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:253
253,747,889/Historical Note

== Historical Note on $253 \, 747 \, 889$ ==
The first case of Fermat's Last Theorem, where the value $p$ in the equation $x^p + y^p = z^p$ does not divide one of $x$, $y$ and $z$, had been proved impossible for all values of $p$ up to $253 \, 747 \, 889$, at the point at which finally proved it in $1994$.
== Sources ==
* : $253,747,889$
Category:Historical Notes
2538

== Number ==
$2538$ (two thousand, five hundred and thirty-eight) is:
:$2 \times 3^3 \times 41$
:The eventual result on applying the modified Kaprekar process on a $4$-digit number
== Also see ==
* Modified Kaprekar Process on 4-Digit Number terminates in 2538
== Sources ==
* : $2538$
Category:Specific Numbers
Category:2538
254

== Number ==
$254$ (two hundred and fifty-four) is:
:$2 \times 127$
:The $40$th nontotient:
::$\nexists m \in \Z_{>0}: \phi \left({m}\right) = 254$
:where $\phi \left({m}\right)$ denotes the Euler $\phi$ function
:The $50$th positive integer $n$ such that no factorial of an integer can end with $n$ zeroes.
== Also see ==
* 
* 
Category:Specific Numbers
Category:254
2542

== Number ==
$2542$ (two thousand, five hundred and forty-two) is:
:$2 \times 31 \times 41$
:The $41$st second pentagonal number after $2$, $7$, $15$, $26$, $\ldots$, $1107$, $1190$, $1276$, $1365$, $1457$, $1552$, $1650$, $1751$, $1855$, $1962$, $2072$, $2185$, $2301$, $2420$:
::$2542 = \dfrac {41 \paren {3 \times 41 + 1} } 2$
:The $82$nd generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $2072$, $2147$, $2185$, $2262$, $2301$, $2380$, $2420$, $2501$:
::$2542 = \dfrac {41 \paren {3 \times 41 + 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:2542
255

== Number ==
$255$ (two hundred and fifty-five) is:
:$3 \times 5 \times 17$
:The $25$th sphenic number after $30$, $42$, $66$, $70$, $\ldots$, $182$, $186$, $190$, $195$, $222$, $230$, $231$, $246$:
::$255 = 3 \times 5 \times 17$
:The $7$th positive integer solution after $1$, $3$, $15$, $104$, $164$, $194$ to $\phi \left({n}\right) = \phi \left({n + 1}\right)$:
::$\phi \left({255}\right) = 128 = \phi \left({256}\right)$
== Also see ==
* 
* 
Category:Specific Numbers
Category:255
2556

== Number ==
$2556$ (two thousand, five hundred and fifty-six) is:
:$2^2 \times 3^2 \times 71$
:The largest term of the amicable triplet $\tuple {1980, 2016, 2556}$:
::$\map {\sigma_1} {1980} = \map {\sigma_1} {2016} = \map {\sigma_1} {2556} = 6552 = 1980 + 2016 + 2556$
:The $36$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $1431$, $1540$, $1653$, $1770$, $1891$, $2016$, $2145$, $2278$, $2415$:
::$2556 = \ds \sum_{k \mathop = 1}^{36} \paren {4 k - 3} = 36 \paren {2 \times 35 - 1}$
:The $71$st triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $1891$, $1953$, $2016$, $2080$, $2145$, $2211$, $2278$, $2346$, $2415$, $2485$:
::$2556 = \ds \sum_{k \mathop = 1}^{71} k = \dfrac {71 \times \paren {71 + 1} } 2$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:2556
256

== Number ==
$256$ (two hundred and fifty-six) is:
:$2^8$
:In binary:
::$10 \, 000 \, 000$
:In hexadecimal:
::$100$
:The $2$nd eighth power after $1$:
::$256 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$
:The $2$nd power of $16$ after $(1)$, $16$:
::$256 = 16^2$
:The $3$rd positive integer after $128$, $192$ with $7$ or more prime factors:
::$256 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \paren {\times \, 2}$
:The $3$rd (and possibly last) power of $2$ after $1$, $4$ which is the sum of distinct powers of $3$:
::$256 = 2^8 = 3^0 + 3^1 + 3^2 + 3^5$
:The $4$th fourth power after $1$, $16$, $81$:
::$256 = 4 \times 4 \times 4 \times 4$
:The $4$th power of $4$ after $(1)$, $4$, $16$, $64$:
::$256 = 4^4$
:The $6$th square number after $1$, $4$, $36$, $121$, $144$ to be the divisor sum value of some (strictly) positive integer:
::$256 = \map {\sigma_1} {217}$
:The $8$th power of $2$ after $(1)$, $2$, $4$, $8$, $16$, $32$, $64$, $128$:
::$256 = 2^8$
:The $9$th almost perfect number after $1$, $2$, $4$, $8$, $16$, $32$, $64$, $128$:
::$\map {\sigma_1} {256} = 511 = 2 \times 256 - 1$
:The $10$th positive integer after $64$, $96$, $128$, $144$, $160$, $192$, $216$, $224$, $240$ with $6$ or more prime factors:
::$256 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \paren {\times \, 2 \times 2}$
:The $16$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $49$, $64$, $81$, $100$, $121$, $144$, $169$, $196$, $225$:
::$256 = 16 \times 16$
:The $27$th powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $\ldots$, $144$, $169$, $196$, $200$, $216$, $225$, $243$
:The number of standard instances of the categorical syllogism
== Also see ==
* Number of Standard Instances of Categorical Syllogism
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
== Sources ==
* : $256$
* : $256$
Category:Specific Numbers
Category:256
256,103,393

== Number ==
$256 \, 103 \, 393$ is:
:$7 \times 13 \times 1181 \times 2383$
:A positive integer which can be expressed as the sum of $3$ fourth powers in $2$ different ways:
:such that:
== Also see ==
* Integer which is Sum of 3 Fourth Powers in 2 Ways and Products of Those Roots
== Sources ==
* : $256,103,393$
Category:Specific Numbers
Category:256,103,393
257

== Number ==
$257$ (two hundred and fifty-seven) is:
:The $55$th prime number
:The $3$rd (and largest known) prime Sierpiński number of the first kind after $2$, $5$:
::$257 = 4^4 + 1$
:The $4$th Fermat number and Fermat prime after $3$, $5$, $17$:
::$257 = 2^{\paren {2^3} } + 1 = 2^8 + 1$
:The $4$th Sierpiński number of the first kind after $2$, $5$, $28$:
::$257 = 4^4 + 1$
:The $6$th balanced prime after $5$, $53$, $157$, $173$, $211$:
::$257 = \dfrac {251 + 263} 2$
:The $7$th prime number of the form $n^2 + 1$ after $2$, $5$, $17$, $37$, $101$, $197$:
::$257 = 16^2 + 1$
:The $10$th Proth prime after $3$, $5$, $13$, $17$, $41$, $97$, $113$, $193$, $241$:
::$257 = 1 \times 2^8 + 1$
:The index of the $12$th and last Mersenne number after $1$, $2$, $3$, $5$, $7$, $13$, $17$, $19$, $31$, $67$, $127$ which asserted to be prime
::(in this case he was not correct: in $1922$, proved that $M_{257}$ is composite)
:The $21$st long period prime after $7$, $17$, $19$, $23$, $29$, $\ldots$, $181$, $193$, $223$, $229$, $233$
== Also see ==
* Construction of Regular $257$-Gon
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
== Sources ==
* : $257$
* : $257$
Category:Specific Numbers
Category:257
258

== Number ==
$258$ (two hundred and fifty-eight) is:
:$2 \times 3 \times 43$
:The $26$th sphenic number after $30$, $42$, $66$, $70$, $\ldots$, $182$, $186$, $190$, $195$, $222$, $230$, $231$, $246$, $255$:
::$258 = 2 \times 3 \times 43$
:The $41$st nontotient:
::$\nexists m \in \Z_{>0}: \phi \left({m}\right) = 258$
:where $\phi \left({m}\right)$ denotes the Euler $\phi$ function
== Also see ==
* 
* 
Category:Specific Numbers
Category:258
258,474,216

== Number ==
$258 \, 474 \, 216$ is:
:$2^3 \times 3 \times 7^2 \times 11 \times 13 \times 29 \times 53$
:The $6$th and largest triangular number after $6$, $120$, $210$, $990$, $185 \, 136$ which can be expressed as the product of $3$ consecutive integers:
::$258 \, 474 \, 216 = T_{22 \, 736} = 636 \times 637 \times 638$
:The $22 \, 736$th triangular number:
::$258 \, 474 \, 216 = \ds \sum_{k \mathop = 1}^{22 \, 736} k = \dfrac {22 \, 736 \times \paren {22 \, 736 + 1} } 2$
== Also see ==
* 
* 
== Sources ==
* : $258,474,216$
Category:Specific Numbers
Category:258,474,216
258,741

== Number ==
$258 \, 741$ (two hundred and fifty-eight thousand, seven hundred and forty-one) is:
:$3^3 \times 7 \times 37^2$
:Equal to $333 \times 777$, and an anagram of $142 \, 857$
== Sources ==
* : $147,852$
* : $147,852$
Category:Specific Numbers
Category:258,741