nyuuzyou/wikis huggingface dataset:
233

== Number ==
$233$ (two hundred and thirty-three) is:
:The $51$st prime number
:The $2$nd element of the smallest triple of consecutive positive integers each of which is the sum of two squares:
::$232 = 14^2 + 6^2$, $233 = 13^2 + 8^2$, $234 = 15^2 + 3^2$
:The $6$th Fibonacci prime after $2$, $3$, $5$, $13$, $89$
:The $12$th near-repdigit prime after $101$, $113$, $131$, $151$, $181$, $191$, $199$, $211$, $223$, $227$, $229$
:The $13$th Fibonacci number, after $1$, $1$, $2$, $3$, $5$, $8$, $13$, $21$, $34$, $55$, $89$, $144$:
::$233 = 89 + 144$
:The $14$th right-truncatable prime after $2$, $3$, $5$, $7$, $23$, $29$, $31$, $37$, $53$, $59$, $71$, $73$, $79$
:The $16$th Sophie Germain prime after $2$, $3$, $5$, $11$, $23$, $29$, $41$, $53$, $83$, $89$, $113$, $131$, $173$, $179$, $191$:
::$2 \times 233 + 1 = 467$, which is prime.
:The $20$th long period prime after $7$, $17$, $19$, $23$, $29$, $47$, $59$, $61$, $97$, $109$, $113$, $131$, $149$, $167$, $179$, $181$, $193$, $223$, $229$
== Also see ==
* 
* 
* 
* 
* 
* 
* 
* 
== Sources ==
* : $232$
* : $232$
Category:Specific Numbers
Category:233
234

== Number ==
$234$ (two hundred and thirty-four) is:
:$2 \times 3^2 \times 13$
:The $3$rd element of the smallest triple of consecutive positive integers each of which is the sum of two squares:
::$232 = 14^2 + 6^2$, $233 = 13^2 + 8^2$, $234 = 15^2 + 3^2$
:The $4$th element of the $2$nd set of $4$ positive integers which form an arithmetic sequence which all have the same Euler $\phi$ value:
::$\map \phi {216} = \map \phi {222} = \map \phi {228} = \map \phi {234} = 72$
:The $15$th of the $17$ positive integers for which the value of the Euler $\phi$ function is $72$:
::$73$, $91$, $95$, $111$, $117$, $135$, $146$, $148$, $152$, $182$, $190$, $216$, $222$, $228$, $234$, $252$, $270$
:The $34$th nontotient:
::$\nexists m \in \Z_{>0}: \map \phi m = 234$
:where $\map \phi m$ denotes the Euler $\phi$ function
== Also see ==
* 
* 
* 
* 
== Sources ==
* : $232$
* : $232$
Category:Specific Numbers
Category:234
2346

== Number ==
$2346$ (two thousand, three hundred and forty-six) is:
:$2 \times 3 \times 17 \times 23$
:The $68$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $1891$, $1953$, $2016$, $2080$, $2145$, $2211$, $2278$:
::$2346 = \ds \sum_{k \mathop = 1}^{68} k = \dfrac {68 \times \paren {68 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:2346
235

== Number ==
$235$ (two hundred and thirty-five) is:
:$5 \times 47$
:The $10$th heptagonal number after $1$, $7$, $18$, $34$, $55$, $81$, $112$, $148$, $189$:
::$235 = 1 + 7 + 11 + 16 + 21 + 26 + 31 + 36 + 41 + 46 = \dfrac {10 \left({5 \times 10 - 3}\right)} 2$
:The $45$th lucky number:
::$1$, $3$, $7$, $9$, $13$, $15$, $21$, $\ldots$, $193$, $195$, $201$, $205$, $211$, $219$, $223$, $231$, $235$, $\ldots$
:The $46$th positive integer $n$ such that no factorial of an integer can end with $n$ zeroes.
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:235
2356

== Number ==
$2356$ (two thousand, three hundred and fifty-six) is:
:$2^2 \times 19 \times 31$
:The $31$st heptagonal number after $1$, $7$, $18$, $34$, $55$, $81$, $112$, $\ldots$, $1525$, $1651$, $1782$, $1918$, $2059$, $2205$:
::$2356 = \ds \sum_{k \mathop = 1}^{31} \paren {5 k - 4} = \dfrac {31 \paren {5 \times 31 - 3} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:2356
236

== Number ==
$236$ (two hundred and thirty-six) is:
:$2^2 \times 59$
:The $35$th nontotient:
::$\nexists m \in \Z_{>0}: \map \phi m = 236$
:where $\map \phi m$ denotes the Euler $\phi$ function
:The $38$th happy number after $1$, $7$, $10$, $13$, $19$, $23$, $\ldots$, $130$, $133$, $139$, $167$, $176$, $188$, $190$, $192$, $193$, $203$, $208$, $219$, $226$, $230$:
::$236 \to 2^2 + 3^2 + 6^2 = 4 + 9 + 36 = 49 \to 4^2 + 9^2 = 16 + 81 = 97 \to 9^2 + 7^2 = 81 + 49 = 130 \to 1^2 + 3^2 + 0^2 = 1 + 9 + 0 = 10 \to 1^2 + 0^2 = 1$
== Also see ==
* 
* 
Category:Specific Numbers
Category:236
237

== Number ==
$237$ (two hundred and thirty-seven) is:
:$3 \times 79$
:The $46$th lucky number:
::$1$, $3$, $7$, $9$, $13$, $15$, $21$, $\ldots$, $193$, $195$, $201$, $205$, $211$, $219$, $223$, $231$, $235$, $237$, $\ldots$
== Also see ==
* 
Category:Specific Numbers
Category:237
2377

== Number ==
$2377$ (two thousand, three hundred and seventy-seven) is:
:The $352$nd 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:
::$3$, $5$, $11$, $13$, $41$, $89$, $317$, $337$, $991$, $1873$, $2053$, $2377$
== Also see ==
* 
* 
Category:Specific Numbers
Category:2377
238

== Number ==
$238$ (two hundred and thirty-eight) is:
:$2 \times 7 \times 17$
:The $14$th untouchable number after $2$, $5$, $52$, $88$, $96$, $120$, $124$, $146$, $162$, $188$, $206$, $210$, $216$
:The smallest positive integer which can be expressed as the sum of $2$ distinct lucky numbers in $15$ different ways
:The $23$rd sphenic number after $30$, $42$, $66$, $70$, $78$, $102$, $105$, $110$, $114$, $130$, $138$, $154$, $165$, $170$, $174$, $182$, $186$, $190$, $195$, $222$, $230$, $231$:
::$238 = 2 \times 7 \times 17$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:238
2380

== Number ==
$2380$ (two thousand, three hundred and eighty) is:
:$2^2 \times 5 \times 7 \times 17$
:The $14$th pentatope number after $1$, $5$, $15$, $35$, $70$, $126$, $210$, $330$, $495$, $715$, $1001$, $1365$, $1820$:
::$2380 = \ds \sum_{k \mathop = 1}^{14} \dfrac {k \paren {k + 1} \paren {k + 2} } 6 = \dfrac {14 \paren {14 + 1} \paren {14 + 2} \paren {14 + 3} } {24}$
:The $40$th pentagonal number after $1$, $5$, $12$, $22$, $35$, $\ldots$, $1820$, $1926$, $2035$, $2147$, $2262$:
::$2380 = \ds\sum_{k \mathop = 1}^{40} \paren {3 k - 2} = \dfrac {40 \paren {3 \times 40 - 1} } 2$
:The $79$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $1617$, $1650$, $1717$, $1751$, $1820$, $1855$, $1926$, $1962$, $2072$, $2147$, $2185$, $2262$, $2301$:
::$2380 = \ds \sum_{k \mathop = 1}^{40} \paren {3 k - 2} = \dfrac {40 \paren {3 \times 40 - 1} } 2$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:2380
239

== Number ==
$239$ (two hundred and thirty-nine) is:
:The $52$nd prime number
:The $2$nd of the only two positive integers needing as many as $9$ positive cubes to express it:
::$239 = 4^3 + 4^3 + 3^3 + 3^3 + 3^3 + 3^3 + 1^3 + 1^3 + 1^3$
:The other is $23$.
:The $3$rd positive integer after $79$, $159$ which cannot be expressed as the sum of fewer than $19$ fourth powers:
::$239 = 13 \times 1^4 + 4 \times 2^4 + 2 \times 3^4$
:The smallest positive integer the decimal expansion of whose reciprocal has a period of $7$:
::$\dfrac 1 {239} = 0 \cdotp \dot 00418 \, 4 \dot 1$
:The $15$th right-truncatable prime after $2$, $3$, $5$, $7$, $23$, $29$, $31$, $37$, $53$, $59$, $71$, $73$, $79$, $233$
:The smaller of the $17$th pair of twin primes, with $241$
:The $17$th Sophie Germain prime after $2$, $3$, $5$, $11$, $23$, $29$, $41$, $53$, $83$, $89$, $113$, $131$, $173$, $179$, $191$, $233$:
::$2 \times 239 + 1 = 479$, which is prime.
:The $39$th happy number after $1$, $7$, $10$, $13$, $19$, $23$, $\ldots$, $188$, $190$, $192$, $193$, $203$, $208$, $219$, $226$, $230$, $236$:
::$239 \to 2^2 + 3^2 + 9^2 = 4 + 9 + 81 = 94 \to 9^2 + 4^2 = 81 + 16 = 97 \to 9^2 + 7^2 = 81 + 49 = 130 \to 1^2 + 3^2 + 0^2 = 1 + 9 + 0 = 10 \to 1^2 + 0^2 = 1$
:The largest integer such that the largest prime factor of $n^2 + 1$ is less than $17$:
::$239^2 + 1 = 57122 = 2 \times 13^4$
:With $y = 13$, the only $x$ which is the solution of the indeterminate Diophantine equation $x^2 + 1 = 2 y^4$:
::$239^2 + 1 = 2 \times 13^4$
== Also see ==
* Numbers not Expressible as Sum of Less than 9 Positive Cubes
* Largest Prime Factor of n squared plus 1
* Solution of Diophantine Equation $x^2 + 1 = 2 y^4$
* 
* 
* 
* 
* 
* 
* 
== Sources ==
* : $23$
* : $239$
* : $23$
* : $239$
Category:Specific Numbers
Category:239
2397

== Number ==
$2397$ (two thousand, three hundred and ninety-seven) is:
:$3 \times 17 \times 47$
:The $10$th after $4$, $13$, $38$, $87$, $208$, $377$, $666$, $1027$, $1556$ in the sequence formed by adding the squares of the first $n$ primes:
::$2397 = \ds \sum_{i \mathop = 1}^9 {p_i}^2 = 2^2 + 3^2 + 5^2 + 7^2 + 11^2 + 13^2 + 17^2 + 19^2 + 23^2 + 29^2$
== Also see ==
* 
Category:Specific Numbers
Category:2397
239 is not Expressible as Sum of Fewer than 19 Fourth Powers

== Theorem ==
$239$ cannot be expressed as the sum of fewer than $19$ fourth powers:
:$239 = 13 \times 1^4 + 4 \times 2^4 + 2 \times 3^4$
== Proof ==
First note that $4^4 = 256 > 239$.
Then note that $3 \times 3^4 = 243 > 239$.
Hence any expression of $239$ as fourth powers uses no $n^4$ for $n \ge 4$, and uses not more than $2$ instances $3^4$.
For the remainder, using $2^4$ uses fewer fourth powers than $16$ instances $1^4$ does.
Now we have:
and it is apparent the first one uses the least number of fourth powers, at $19$.
== Also see ==
* Hilbert-Waring Theorem
* Numbers not Expressible as Sum of Fewer than 19 Fourth Powers
== Sources ==
* : $239$
* : $239$
Category:Fourth Powers
Category:Hilbert-Waring Theorem
Category:239
23 is Largest Integer not Sum of Distinct Perfect Powers

== Theorem ==
The largest integer which cannot be expressed as the sum of distinct perfect powers is $23$.
== Proof ==
The first few perfect powers are:
:$1, 4, 8, 9, 16, 25, 27, 32, \dots$
First we show that $23$ cannot be expressed as the sum of distinct perfect powers.
Only $1, 4, 8, 9, 16$ are perfect powers less than $23$.
Suppose $23$ can be so expressed.
Since $1 + 4 + 8 + 9 = 22 < 23$, $16$ must be used in the sum.
However $23 - 16 = 7$ cannot be expressed as a sum of $1$ and $4$.
Thus $23$ cannot be expressed as the sum of distinct perfect powers.
Now we show that all numbers greater than $23$ can be so expressed.
By Richert's Theorem, we just need to show:
:For any $23 < n \le 23 + 32$, $n$ can be expressed as a sum of distinct elements in $\set {1, 4, 8, 9, 16, 25, 27}$
:$s_{i + 1} \le 2 s_i$ for every $i \ge 7$, where $s_i$ is the $i$th perfect power
Verification of the first statement is included in the bottom of this proof.
To verify the second statement:
Let $i \ge 7$.
Let $m$ be the integer satisfying:
:$2^{m + 1} > s_i \ge 2^m$
Note that $2^{m + 1}$ is also a perfect power.
There must be a perfect power greater than $s_i$ but not greater than $2^{m + 1}$.
Thus:
:$s_{i + 1} \le 2^{m + 1} = 2 \cdot 2^m \le 2 s_i$
Therefore $23$ is the largest integer that cannot be expressed as the sum of distinct perfect powers..
Here is $23 < n \le 55$ expressed as a sum of distinct elements in $\set {1, 4, 8, 9, 16, 25, 27}$:
== Sources ==
* : $23$
* : $23$
Category:23
Category:Powers
23 is Largest Integer not Sum of Distinct Perfect Powers/Mistake

== Source Work ==
:
:The Dictionary
::$23$
:
:The Dictionary
::$23$
== Mistake ==
:$23$ is the largest integer that is not the sum of distinct powers.
== Correction ==
This statement is refuted by:
:$23 2^4 + 2^2 + 2^1 + 2^0$
What in fact is meant is:
:$23$ is the largest integer that is not the sum of distinct perfect powers.
== Sources ==
* : $23$
* : $23$
Category:Mistakes/Curious and Interesting Numbers
24,029

== Number ==
$24 \, 029$ (twenty-four thousand and twenty-nine) is:
:The $2673$rd prime number
:The $18$th prime $p$ such that $p \# + 1$, where $p \#$ denotes primorial (product of all primes up to $p$) of $p$, is prime:
::$2$, $3$, $5$, $7$, $11$, $31$, $379$, $1019$, $1021$, $2657$, $3229$, $4547$, $4787$, $11 \, 549$, $13 \, 649$, $18 \, 523$, $23 \, 801$, $24 \, 029$
== Also see ==
* 
* 
== Historical Note ==
== Sources ==
* 
* : $24,029 \# + 1$
Category:Specific Numbers
Category:24,029
24,678,050

== Number ==
$24 \, 678 \, 050$ is:
:$2 \times 5^2 \times 17 \times 29 \, 033$
:The $25$th pluperfect digital invariant after $1$, $2$, $3$, $4$, $5$, $6$, $\ldots$, $93 \, 084$, $548 \, 834$, $1 \, 741 \, 725$, $4 \, 210 \, 818$, $9 \, 800 \, 817$, $9 \, 926 \, 315$:
== Also see ==
* 
== Sources ==
* : $24,678,050$
* : $24,678,050$
Category:Specific Numbers
Category:24,678,050
24,678,051

== Number ==
$24 \, 678 \, 051$ is:
:$3 \times 8 \, 226 \, 017$
:The $26$th pluperfect digital invariant after $1$, $2$, $3$, $4$, $5$, $6$, $\ldots$, $1 \, 741 \, 725$, $4 \, 210 \, 818$, $9 \, 800 \, 817$, $9 \, 926 \, 315$, $24 \, 678 \, 050$:
== Also see ==
* 
Category:Specific Numbers
Category:24,678,051
24,883,200

== Number ==
$24 \, 883 \, 200$ (twenty-four million, eight hundred and eighty-three thousand, two hundred) is:
:$2^{12} \times 3^5 \times 5^2$
:The $6$th superfactorial after $1$, $2$, $12$, $288$, $34 \, 560$:
::$24 \, 883 \, 200 = 6\$ = 6! \times 5! \times 4! \times 3! \times 2! \times 1!$
== Also see ==
* 
Category:Specific Numbers
Category:24,883,200
24/Historical Note

== Historical Note on $24$ (Twenty-Four) ==
$24$ occurs frequently as a subdivision of a standard measure:
:There are $24$ hours in $1$ day.
:There are $24$ scruples in $1$ apothecaries' ounce.
:There are $24$ grains in $1$ pennyweight.
== Sources ==
* : $24$
* : $24$
Category:Historical Notes
240

== Number ==
$240$ (two hundred and forty) is:
:$2^4 \times 3 \times 5$
:The length of the longest edge of the smallest cuboid whose edges and the diagonals of whose faces are all integers:
::The lengths of the edges are $44, 117, 240$
::The lengths of the diagonals of the faces are $125, 244, 267$.
:The $9$th positive integer after $64$, $96$, $128$, $144$, $160$, $192$, $216$, $224$ with $6$ or more prime factors:
::$240 = 2 \times 2 \times 2 \times 2 \times 3 \times 5$
:The $12$th highly composite number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$, $60$, $120$, $180$:
::$\map {\sigma_0} {240} = 20$
:The $12$th superabundant number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$, $60$, $120$, $180$:
::$\dfrac {\map {\sigma_1} {240} } {240} = \dfrac {744} {240} = 3 \cdotp 1$
:The $29$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $12$, $16$, $18$, $20$, $24$, $30$, $36$, $42$, $48$, $60$, $72$, $84$, $90$, $96$, $108$, $120$, $144$, $168$, $180$, $210$, $216$:
::$\map {\sigma_1} {240} = 744$
:An integer with over $240$ divisors is greater than $1 \, 000 \, 000$.
== Also see ==
* Cuboid with Integer Edges and Face Diagonals
* Number with over 240 Divisors is greater than 1,000,000
* 
* 
* 
* 
== Sources ==
* : $240$
* : $240$
Category:Specific Numbers
Category:240
2401

== Number ==
$2401$ (two thousand four hundred and one) is:
:$7^4$
:The $1$st $4$th power that can be expressed as the sum of $2$ $4$th powers minus a $3$rd with all numbers less than $10^4$:
::$2401 = 7^4 = 227^4 + 157^4 - 239^4$
:The $4$th power of $7$ after $(1)$, $7$, $49$, $343$:
::$2401 = 7^4$
:The $7$th fourth power after $1$, $16$, $81$, $256$, $625$, $1296$:
::$2401 = 7 \times 7 \times 7 \times 7$
:The $49$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $1296$, $1369$, $1444$, $1521$, $1600$, $1681$, $1764$, $1849$, $1936$, $2025$, $2116$, $2209$, $2304$:
::$2401 = 49 \times 49$
== Also see ==
* Smallest Fourth Power as Sum and Difference of Fourth Powers
* 
* 
* 
== Sources ==
* : $2401$
Category:Specific Numbers
Category:2401
241

== Number ==
$241$ (two hundred and forty-one) is:
:The $53$rd prime number.
:The smaller of the $3$rd pair of primes whose prime gap is $10$:
::$251 - 241 = 10$
:The $9$th Proth prime after $3$, $5$, $13$, $17$, $41$, $97$, $113$, $193$:
::$241 = 15 \times 2^4 + 1$
:The larger of the $17$th pair of twin primes, with $239$
:The $47$th lucky number:
::$1$, $3$, $7$, $9$, $13$, $15$, $21$, $\ldots$, $193$, $195$, $201$, $205$, $211$, $219$, $223$, $231$, $235$, $237$, $241$, $\ldots$
:The $47$th positive integer $n$ such that no factorial of an integer can end with $n$ zeroes.
== Also see ==
* 
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:241
2415

== Number ==
$2415$ (two thousand, four hundred and fifteen) is:
:$3 \times 5 \times 7 \times 23$
:The $35$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $1431$, $1540$, $1653$, $1770$, $1891$, $2016$, $2145$, $2278$:
::$2415 = \ds \sum_{k \mathop = 1}^{35} \paren {4 k - 3} = 35 \paren {2 \times 35 - 1}$
:The $69$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $1891$, $1953$, $2016$, $2080$, $2145$, $2211$, $2278$, $2346$:
::$2415 = \ds \sum_{k \mathop = 1}^{69} k = \dfrac {69 \times \paren {69 + 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:2415
2418

== Number ==
$2418$ (two thousand, four hundred and eighteen) is:
:$2 \times 3 \times 13 \times 31$
:The $11$th positive integer after $504$, $756$, $806$, $1008$, $1148$, $1209$, $1472$, $1512$, $2016$, $2208$ which can be expressed as the product of $2$ two-digit numbers in $2$ ways such that the factors in one of those pairs is the reversal of each of the factors in the other:
::$2208 = 26 \times 93 = 39 \times 62$
== Also see ==
* 
Category:Specific Numbers
Category:2418
242

== Number ==
$242$ (two hundred and forty-two) is:
:$2 \times 11^2$
:The $36$th nontotient:
::$\nexists m \in \Z_{>0}: \map \phi m = 242$
:where $\map \phi m$ denotes the Euler $\phi$ function
:The $9$th integer after $0$, $1$, $2$, $4$, $8$, $121$, $151$, $212$ which is palindromic in both decimal and ternary:
::$242_{10} = 22 \, 222_3$
:The $1$st of the $7$th pair of consecutive integers which both have $6$ divisors:
::$\map {\sigma_0} {242} = \map {\sigma_0} {243} = 6$
:The $1$st of the $1$st quadruple of consecutive integers which all have an equal divisors:
::$\map {\sigma_0} {242} = \map {\sigma_0} {243} = \map {\sigma_0} {244} = \map {\sigma_0} {245} = 6$
== Also see ==
* 
* 
* 
* 
== Sources ==
* : $242$
* : $242$
Category:Specific Numbers
Category:242
2420

== Number ==
$2420$ (two thousand, four hundred and twenty) is:
:$2^2 \times 5 \times 11^2$
:The $40$th second pentagonal number after $2$, $7$, $15$, $26$, $\ldots$, $1751$, $1855$, $1962$, $2072$, $2185$, $2301$:
::$2420 = \dfrac {40 \paren {3 \times 40 + 1} } 2$
:The $80$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $2072$, $2147$, $2185$, $2262$, $2301$, $2380$:
::$2420 = \dfrac {40 \paren {3 \times 40 + 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:2420
2427

== Number ==
$2427$ (two thousand four hundred and twenty-seven) is:
:$3 \times 809$
:The $18$th number after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $89$, $135$, $175$, $518$, $598$, $1306$, $1676$ which is the sum of the increasing powers of its digits taken in order:
::$2^1 + 4^2 + 2^3 + 7^4 = 2427$
== Also see ==
* 
Category:Specific Numbers
Category:2427
243

== Number ==
$243$ (two hundred and forty-three) is:
:$3^5$
:In ternary:
::$100 \, 000_3$
:The $3$rd fifth power after $1$, $32$:
::$243 = 3 \times 3 \times 3 \times 3 \times 3$
:The number of different binary operations with an identity element that can be applied to a set with $3$ elements
:The $5$th power of $3$ after $(1)$, $3$, $9$, $27$, $81$:
::$243 = 3^5$
:The $26$th powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $\ldots$, $108$, $121$, $125$, $128$, $144$, $169$, $196$, $200$, $216$, $225$
:The $2$nd of the $7$th pair of consecutive integers which both have $6$ divisors:
::$\map {\sigma_0} {242} = \map {\sigma_0} {243} = 6$
:The $1$st of the $8$th pair of consecutive integers which both have $6$ divisors:
::$\map {\sigma_0} {243} = \map {\sigma_0} {244} = 6$
:The $2$nd of the $1$st quadruple of consecutive integers which all have an equal divisors:
::$\map {\sigma_0} {242} = \map {\sigma_0} {243} = \map {\sigma_0} {244} = \map {\sigma_0} {245} = 6$
== Also see ==
* 
* 
* 
* 
* 
* 
== Sources ==
* : $242$
* : $243$
* : $242$
* : $243$
Category:Specific Numbers
Category:243
243,760

== Number ==
$243 \, 760$ (two hundred and forty-three thousand, seven hundred and sixty) is:
:$2^4 \times 5 \times 11 \times 277$
:The $16$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:243,760
2437

== Number ==
$2437$ (two thousand, four hundred and thirty-seven) is:
:The $361$st prime number
:The $29$th centered hexagonal number after $1$, $7$, $19$, $37$, $61$, $91$, $127$, $\ldots$, $1387$, $1519$, $1657$, $1801$, $1951$, $2107$, $2269$:
::$2437 = \ds 1 + \sum_{k \mathop = 1}^{29 - 1} 6 k = 29^3 - 28^3$
== Also see ==
* 
* 
Category:Specific Numbers
Category:2437
244

== Number ==
$244$ (two hundred and forty-four) is:
:$2^2 \times 61$
:The $37$th nontotient:
::$\nexists m \in \Z_{>0}: \map \phi m = 244$
:where $\map \phi m$ denotes the Euler $\phi$ function
:The $23$rd noncototient after $10$, $26$, $34$, $50$, $52$, $58$, $\ldots$, $170$, $172$, $186$, $202$, $206$, $218$, $222$, $232$:
::$\nexists m \in \Z_{>0}: m - \map \phi m = 244$
:where $\map \phi m$ denotes the Euler $\phi$ function
:The $2$nd of the $8$th pair of consecutive integers which both have $6$ divisors:
::$\map {\sigma_0} {243} = \map {\sigma_0} {245} = 6$
:The $1$st of the $9$th pair of consecutive integers which both have $6$ divisors:
::$\map {\sigma_0} {244} = \map {\sigma_0} {245} = 6$
:The $3$rd of the $1$st quadruple of consecutive integers which all have an equal divisors:
::$\map {\sigma_0} {242} = \map {\sigma_0} {243} = \map {\sigma_0} {244} = \map {\sigma_0} {245} = 6$
:The length of the $2$nd longest face diagonal of the smallest cuboid whose edges and the diagonals of whose faces are all integers:
::The lengths of the edges are $44, 117, 240$
::The lengths of the diagonals of the faces are $125, 244, 267$.
== Also see ==
* Cuboid with Integer Edges and Face Diagonals
* 
* 
* 
* 
== Sources ==
* : $242$
* : $242$
Category:Specific Numbers
Category:244
244,197,000,982,499,715,087,866,346

== Number ==
$244 \, 197 \, 000 \, 982 \, 499 \, 715 \, 087 \, 866 \, 346$ is:
:$2 \times 3 \times 11 \times 23 \times 31 \times 47 \, 137 \times 28 \, 282 \, 147 \times 3 \, 892 \, 535 \, 183$
:The $10$th Giuga number:
::$\dfrac 1 2 + \dfrac 1 3 + \dfrac 1 {11} + \dfrac 1 {23} + \dfrac 1 {31} + \dfrac 1 {47 \, 137} + \dfrac 1 {28 \, 282 \, 147} + \dfrac 1 {3 \, 892 \, 535 \, 183} - \dfrac 1 {244 \, 197 \, 000 \, 982 \, 499 \, 715 \, 087 \, 866 \, 346} = 1$
== Also see ==
* 
== Sources ==
* : $244,197,000,982,499,715,087,866,346$
Category:Specific Numbers
Category:244,197,000,982,499,715,087,866,346
245

== Number ==
$245$ (two hundred and forty-five) is:
:$5 \times 7^2$
:The $2$nd of the $9$th pair of consecutive integers which both have $6$ divisors:
::$\map {\sigma_0} {244} = \map {\sigma_0} {245} = 6$
:The $4$th of the $1$st quadruple of consecutive integers which all have an equal divisors:
::$\map {\sigma_0} {242} = \map {\sigma_0} {243} = \map {\sigma_0} {244} = \map {\sigma_0} {245} = 6$
:The largest odd positive integer that cannot be expressed as the sum of exactly $5$ non-zero square numbers all of which are coprime.
:The $16$th positive integer $n$ after $0$, $1$, $5$, $25$, $29$, $41$, $49$, $61$, $65$, $85$, $89$, $101$, $125$, $145$, $149$ such that the Fibonacci number $F_n$ ends in $n$
== Also see ==
* Odd Numbers Not Expressible as Sum of 5 Distinct Non-Zero Coprime Squares
* 
* 
* 
== Sources ==
* : $242$
* : $157$
* : $242$
Category:Specific Numbers
Category:245
246

== Number ==
$246$ (two hundred and forty-six) is:
:$2 \times 3 \times 41$
:The $15$th untouchable number after $2$, $5$, $52$, $88$, $96$, $120$, $124$, $146$, $162$, $188$, $206$, $210$, $216$, $238$
:The $24$th sphenic number after $30$, $42$, $66$, $70$, $78$, $102$, $105$, $110$, $114$, $130$, $138$, $154$, $165$, $170$, $174$, $182$, $186$, $190$, $195$, $222$, $230$, $231$:
::$246 = 2 \times 3 \times 41$
:The $38$th nontotient:
::$\nexists m \in \Z_{>0}: \map \phi m = 246$
:where $\map \phi m$ denotes the Euler $\phi$ function
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:246