nyuuzyou/wikis huggingface dataset:
2207

== Number ==
$2207$ (two thousand, two hundred and seven) is:
:The $329$th prime number
:The $9$th Lucas prime after $2$, $3$, $7$, $11$, $29$, $47$, $199$, $521$.
:The $16$th Lucas number after $(2)$, $1$, $3$, $4$, $7$, $11$, $18$, $29$, $47$, $76$, $123$, $199$, $322$, $521$, $843$, $1364$:
::$2207 = 843 + 1364$
== Also see ==
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Category:Specific Numbers
Category:2207
2208

== Number ==
$2208$ (two thousand, two hundred and eight) is:
:$2^5 \times 3 \times 23$
:The $11$th Keith number after $14$, $19$, $28$, $47$, $61$, $75$, $197$, $742$, $1104$, $1537$:
::$2$, $2$, $0$, $8$, $12$, $22$, $42$, $84$, $160$, $308$, $594$, $1146$, $2208$, $\ldots$
:The $10$th positive integer after $504$, $756$, $806$, $1008$, $1148$, $1209$, $1472$, $1512$, $2016$ 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:
::$2208 = 23 \times 96 = 69 \times 32$
== Also see ==
* 
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Category:Specific Numbers
Category:2208
2209

== Number ==
$2209$ (two thousand, two hundred and nine) is:
:$47^2$
:The $41$st positive integer which cannot be expressed as the sum of a square and a prime:
::$1$, $10$, $25$, $34$, $58$, $64$, $85$, $\ldots$, $1024$, $1089$, $1225$, $1255$, $1351$, $1414$, $1444$, $1521$, $1681$, $1849$, $1906$, $1936$, $2116$, $2209$, $\ldots$
:The $47$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $1296$, $1369$, $1444$, $1521$, $1600$, $1681$, $1764$, $1849$, $1936$, $2025$, $2116$:
::$2209 = 47 \times 47$
== Also see ==
* 
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Category:Specific Numbers
Category:2209
221

== Number ==
$221$ (two hundred and twenty-one) is:
:$13 \times 17$
:The $11$th positive integer after $50$, $65$, $85$, $125$, $130$, $145$, $170$, $185$, $200$, $205$ which can be expressed as the sum of two square numbers in two or more different ways:
::$221 = 14^2 + 5^2 = 11^2 + 10^2$
== Also see ==
* 
Category:Specific Numbers
Category:221
2211

== Number ==
$2211$ (two thousand, two hundred and eleven) is:
:$3 \times 11 \times 67$
:The $66$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $1326$, $1378$, $1431$, $1485$, $1540$, $1596$, $1653$, $1711$, $1770$, $1770$, $1830$, $1891$, $1953$, $2016$, $2080$, $2145$:
::$2211 = \ds \sum_{k \mathop = 1}^{66} k = \dfrac {66 \times \paren {66 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:2211
222

== Number ==
$222$ (two hundred and twenty-two) is:
:$2 \times 3 \times 37$
:The $2$nd 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 $12$th second pentagonal number after $2$, $7$, $15$, $26$, $40$, $57$, $77$, $100$, $126$, $155$, $187$:
::$222 = \dfrac {12 \paren {3 \times 12 + 1} } 2$
:The $13$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 $20$th sphenic number after $30$, $42$, $66$, $70$, $78$, $102$, $105$, $110$, $114$, $130$, $138$, $154$, $165$, $170$, $174$, $182$, $186$, $190$, $195$:
::$222 = 2 \times 3 \times 37$
:The $21$st noncototient after $10$, $26$, $34$, $50$, $52$, $58$, $86$, $100$, $116$, $122$, $130$, $134$, $146$, $154$, $170$, $172$, $186$, $202$, $206$, $218$:
::$\nexists m \in \Z_{>0}: m - \map \phi m = 222$
:where $\map \phi m$ denotes the Euler $\phi$ function
:The $24$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $100$, $117$, $126$, $145$, $155$, $176$, $187$, $210$:
::$222 = \dfrac {12 \paren {3 \times 12 + 1} } 2$
== Also see ==
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Category:Specific Numbers
Category:222
2223

== Number ==
$2223$ (two thousand, two hundred and twenty-three) is:
:$3^2 \times 13 \times 19$
:The $9$th Kaprekar number after $1$, $9$, $45$, $55$, $99$, $297$, $703$, $999$:
::$2223^2 = 4 \, 941 \, 729 \to 494 + 1729 = 2223$
== Also see ==
* 
Category:Specific Numbers
Category:2223
223

== Number ==
$223$ (two hundred and twenty-three) is:
:The $48$th prime number
:The largest positive integer which cannot be expressed as the sum of less than $37$ positive fifth powers:
::$223 = 31 \times 1^5 + 6 \times 2^5$
:The $9$th near-repdigit prime after $101$, $113$, $131$, $151$, $181$, $191$, $199$, $211$
:The $14$th integer after $7$, $13$, $19$, $35$, $38$, $41$, $57$, $65$, $70$, $125$, $130$, $190$, $205$ the decimal representation of whose square can be split into two parts which are each themselves square:
::$223^2 = 49 \, 729$; $49 = 7^2$, $729 = 27^2$
:The $18$th long period prime after $7$, $17$, $19$, $23$, $29$, $47$, $59$, $61$, $97$, $109$, $113$, $131$, $149$, $167$, $179$, $181$, $193$
:The $21$st left-truncatable prime after $2$, $3$, $5$, $7$, $13$, $17$, $23$, $37$, $43$, $47$, $53$, $67$, $73$, $83$, $97$, $113$, $137$, $167$, $173$, $197$
:The $43$rd lucky number:
::$1$, $3$, $7$, $9$, $13$, $15$, $21$, $\ldots$, $163$, $169$, $171$, $189$, $193$, $195$, $201$, $205$, $211$, $219$, $223$, $\ldots$
:The $44$th positive integer $n$ such that no factorial of an integer can end with $n$ zeroes.
== Also see ==
* Largest Number not Expressible as Sum of Less than 37 Positive Fifth Powers
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== Sources ==
* : $223$
Category:Specific Numbers
Category:223
224

== Number ==
$224$ (two hundred and twenty-four) is:
:$2^5 \times 7$
:The $3$rd element of the $1$st set of $3$ integers $T$ such that $m \map {\sigma_0} m$ is equal for each $m \in T$:
::$168 \times \map {\sigma_0} {168} = 192 \times \map {\sigma_0} {192} = 224 \times \map {\sigma_0} {224} = 2688$
:The $8$th positive integer after $64$, $96$, $128$, $144$, $160$, $192$, $216$ with $6$ or more prime factors:
::$224 = 2 \times 2 \times 2 \times 2 \times 2 \times 7$
:The $25$th Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$, $12$, $15$, $24$, $36$, $111$, $112$, $115$, $128$, $132$, $135$, $144$, $175$, $212$, $216$:
::$224 = 14 \times 16 = 14 \times \paren {2 \times 2 \times 4}$
== Also see ==
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== Sources ==
* : $168$
Category:Specific Numbers
Category:224
225

== Number ==
$225$ (two hundred and twenty-five) is:
:$2^2 \times 3^2$
:The $2$nd power of $15$ after $(1)$, $15$:
::$225 = 15^2$
:The sum of the first $5$ cubes:
::$225 = 1^3 + 2^3 + 3^3 + 4^3 + 5^3$
:The $9$th octagonal number, after $1$, $8$, $21$, $40$, $65$, $96$, $133$, $176$:
::$225 = 1 + 7 + 13 + 19 + 25 + 31 + 37 + 43 + 49 = 9 \paren {3 \times 9 - 2}$
:The $12$th square after $1$, $4$, $9$, $16$, $25$, $36$, $49$, $64$, $81$, $121$, $144$ which has no more than $2$ distinct digits and does not end in $0$:
::$225 = 15^2$
:The $15$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $49$, $64$, $81$, $100$, $121$, $144$, $169$, $196$:
::$225 = 15 \times 15$
:The $25$th powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $\ldots$, $108$, $121$, $125$, $128$, $144$, $169$, $196$, $200$, $216$
== Also see ==
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Category:Specific Numbers
Category:225
226

== Number ==
$226$ (two hundred and twenty-six) is:
:$2 \times 113$
:The $8$th non-square positive integer which cannot be expressed as the sum of a square and a prime:
::$10$, $34$, $58$, $85$, $91$, $130$, $214$, $226$, $\ldots$
:The $14$th positive integer which cannot be expressed as the sum of a square and a prime:
::$1$, $10$, $25$, $34$, $58$, $64$, $85$, $91$, $121$, $130$, $169$, $196$, $214$, $226$, $\ldots$
:The smallest positive integer which can be expressed as the sum of $2$ distinct lucky numbers in $14$ different ways
:The $36$th happy number after $1$, $7$, $10$, $13$, $19$, $23$, $\ldots$, $130$, $133$, $139$, $167$, $176$, $188$, $190$, $192$, $193$, $203$, $208$, $219$:
::$226 \to 2^2 + 2^2 + 6^2 = 4 + 4 + 36 = 44 \to 4^2 + 4^2 = 16 + 16 = 32 \to 3^2 + 2^2 = 9 + 4 = 13 \to 1^2 + 3^2 = 9 + 1 = 10 \to 1^2 + 0^2 = 1$
== Also see ==
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Category:Specific Numbers
Category:226
2262

== Number ==
$2262$ (two thousand, two hundred and sixty-two) is:
:$2 \times 29 \times 39$
:The $39$th pentagonal number after $1$, $5$, $12$, $22$, $35$, $\ldots$, $925$, $1001$, $1080$, $1162$, $1247$, $1335$, $1426$, $1520$, $1617$, $1717$, $1820$, $1926$, $2035$, $2147$:
::$2262 = \ds \sum_{k \mathop = 1}^{39} \paren {3 k - 2} = \dfrac {39 \paren {3 \times 39 - 1} } 2$
:The $77$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $1617$, $1650$, $1717$, $1751$, $1820$, $1855$, $1926$, $1962$, $2072$, $2147$, $2185$:
::$2262 = \ds \sum_{k \mathop = 1}^{39} \paren {3 k - 2} = \dfrac {39 \paren {3 \times 39 - 1} } 2$
== Also see ==
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Category:Specific Numbers
Category:2262
2269

== Number ==
$2269$ (two thousand, two hundred and sixty-nine) is:
:The $337$th prime number
:The $28$th centered hexagonal number after $1$, $7$, $19$, $37$, $61$, $91$, $127$, $\ldots$, $1387$, $1519$, $1657$, $1801$, $1951$, $2107$:
::$2269 = \ds 1 + \sum_{k \mathop = 1}^{28 - 1} 6 k = 28^3 - 27^3$
== Also see ==
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Category:Specific Numbers
Category:2269
227

== Number ==
$227$ (two hundred and twenty-seven) is:
:The $49$th prime number
:The $5$th Stern number after $1$, $3$, $17$, $137$
:The $5$th Stern prime after $1$, $3$, $17$, $137$
:The $10$th near-repdigit prime after $101$, $113$, $131$, $151$, $181$, $191$, $199$, $211$, $223$
:The $11$th safe prime after $5$, $7$, $11$, $23$, $47$, $59$, $83$, $107$, $167$, $179$:
::$227 = 2 \times 113 + 1$
:The smaller of the $16$th pair of twin primes, with $229$
== Also see ==
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Category:Specific Numbers
Category:227
2278

== Number ==
$2278$ (two thousand, two hundred and seventy-eight) is:
:$2 \times 17 \times 67$
:The $34$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $703$, $780$, $861$, $946$, $1035$, $1225$, $1326$, $1431$, $1540$, $1653$, $1770$, $1891$, $2016$, $2145$:
::$2278 = \ds \sum_{k \mathop = 1}^{34} \paren {4 k - 3} = 34 \paren {2 \times 34 - 1}$
:The $67$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $1326$, $1378$, $1431$, $1485$, $1540$, $1596$, $1653$, $1711$, $1770$, $1770$, $1830$, $1891$, $1953$, $2016$, $2080$, $2145$, $2211$:
::$2278 = \ds \sum_{k \mathop = 1}^{67} k = \dfrac {67 \times \paren {67 + 1} } 2$
== Also see ==
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Category:Specific Numbers
Category:2278
228

== Number ==
$228$ (two hundred and twenty-eight) is:
:$2^2 \times 3 \times 19$
:The $2$nd integer after $3$ whose square is the sum of $2$ coprime cubes:
::$228^2 = 37^3 + 11^3$
:The $3$rd 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 $14$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 ==
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== Sources ==
* : $51,984$
Category:Specific Numbers
Category:228
2281

== Number ==
$2281$ (two thousand, two hundred and eighty-one) is:
:The $339$th prime number
:The index of the $17$th Mersenne prime after $2$, $3$, $5$, $7$, $13$, $17$, $19$, $31$, $61$, $89$, $107$, $127$, $521$, $607$, $1279$, $2203$:
::$M_{2281} = 2^{2281} - 1 \approx 4 \cdotp 461 \times 10^{686}$
== Also see ==
* 
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== Sources ==
* : $2^{2281} - 1$
Category:Specific Numbers
Category:2281
2285

== Number ==
$2285$ (two thousand, two hundred and eighty-five) is:
:$5 \times 457$
:The $5$th non-palindromic square root after $26$, $264$, $307$, $836$ of a palindromic square:
::$2285^2 = 5 \, 221 \, 225$
== Also see ==
* 
Category:Specific Numbers
Category:2285
229

== Number ==
$229$ (two hundred and twenty-nine) is:
:The $50$th prime number
:The $11$th of $29$ primes of the form $2 x^2 + 29$:
::$2 \times 10^2 + 29 = 229$ ()
:The $11$th near-repdigit prime after $101$, $113$, $131$, $151$, $181$, $191$, $199$, $211$, $223$, $227$
:The larger of the $16$th pair of twin primes, with $227$
:The $19$th long period prime after $7$, $17$, $19$, $23$, $29$, $47$, $59$, $61$, $97$, $109$, $113$, $131$, $149$, $167$, $179$, $181$, $193$, $223$
:The $45$th positive integer $n$ such that no factorial of an integer can end with $n$ zeroes.
== Also see ==
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Category:Specific Numbers
Category:229
2296

== Number ==
$2296$ (two thousand, two hundred and ninety-six) is:
:$2^3 \times 7 \times 41$
:The $28$th octagonal number, after $1$, $8$, $21$, $40$, $65$, $\ldots$, $645$, $736$, $833$, $936$, $1045$, $1160$, $1281$, $1408$, $1541$, $1680$, $1825$, $1976$, $2133$:
::$2296 = \ds \sum_{k \mathop = 1}^{28} \paren {6 k - 5} = 28 \paren {3 \times 28 - 2}$
== Also see ==
* 
Category:Specific Numbers
Category:2296
23

== Number ==
$23$ (twenty-three) is:
:The $9$th prime number, after $2$, $3$, $5$, $7$, $11$, $13$, $17$, $19$
:The number of unit rods required to brace a unit square when rods may not cross.
:The $1$st of the only two positive integers needing as many as $9$ positive cubes to express it:
::$23 = 2^3 + 2^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3$
:The other is $239$.
:The $1$st element of the $1$st pair of consecutive prime numbers which differ by $6$:
::$23$, $29$: $29 - 23 = 6$
:The $1$st integer greater than $1$ which is not the sum of two Ulam numbers.
:The $2$nd prime $p$ after $11$ such that the Mersenne number $2^p - 1$ is composite
:The $2$nd prime number after $5$ of the form $n! - 1$ for integer $n$:
::$23 = 4! - 1$
:where $n!$ denotes $n$ factorial
:The $3$rd Woodall number after $1$, $7$, and $2$nd Woodall prime after $7$:
::$23 = 3 \times 2^3 - 1$
:The $3$rd Thabit number after $(2)$, $5$, $11$, and $4$th Thabit prime:
::$23 = 3 \times 2^3 - 1$ 
:The index of the $3$rd repunit prime after $R_2$, $R_{19}$:
::$R_{23} = 11 \, 111 \, 111 \, 111 \, 111 \, 111 \, 111 \, 111$
:The $3$rd integer after $1$, $22$ which equals the number of digits in its factorial:
::$23! = 25 \, 852 \, 016 \, 738 \, 884 \, 976 \, 640 \, 000$
:which has $23$ digits
:The $4$th positive integer $n$ after $5$, $11$, $17$ such that no factorial of an integer can end with $n$ zeroes.
:The $4$th of $5$ primes of the form $2 x^2 + 5$:
::$2 \times 3^2 + 5 = 23$ ()
:The $4$th safe prime after $5$, $7$, $11$:
::$23 = 2 \times 11 + 1$
:The $4$th long period prime after $7$, $17$, $19$:
::$\dfrac 1 {23} = 0 \cdotp \dot 04347 \, 82608 \, 69565 \, 21739 \, 1 \dot 3$
:The $5$th Sophie Germain prime after $2$, $3$, $5$, $11$:
::$2 \times 23 + 1 = 47$, which is prime.
:The $5$th right-truncatable prime after $2$, $3$, $5$, $7$
:The $5$th two-sided prime after $2$, $3$, $5$, $7$:
::$23$, $2$, $3$ are prime
:The $5$th prime number after the trivial $2$, $3$, $5$, $7$ consisting of a string of consecutive ascending digits
:The $5$th of $6$ integers after $2$, $5$, $8$, $12$ which cannot be expressed as the sum of distinct triangular numbers
:The $6$th happy number after $1$, $7$, $10$, $13$, $19$:
::$23 \to 2^2 + 3^2 = 4 + 9 = 13 \to 1^2 + 3^2 = 1 + 9 = 10 \to 1^2 + 0^2 = 1$
:The $7$th left-truncatable prime after $2$, $3$, $5$, $7$, $13$, $17$
:The $9$th of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:
::$3$, $4$, $9$, $10$, $12$, $16$, $17$, $22$, $23$, $\ldots$
:The $12$th positive integer which is not the sum of $1$ or more distinct squares:
::$2$, $3$, $6$, $7$, $8$, $11$, $12$, $15$, $18$, $19$, $22$, $23$, $\ldots$
:The $12$th odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
::$1$, $3$, $5$, $7$, $9$, $11$, $13$, $15$, $17$, $19$, $21$, $23$, $\ldots$
== Also see ==
* Numbers not Expressible as Sum of Less than 9 Positive Cubes
* Smallest Integer not Sum of Two Ulam Numbers
* Square-Bracing Problem: Non-Crossing Rods
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== Sources ==
* : $23$
* : $23$
Category:Specific Numbers
Category:23
23,209

== Number ==
$23 \, 209$ (twenty-three thousand, two hundred and nine) is:
:The $2591$st prime number
:The index of the $26$th Mersenne prime after $2$, $3$, $5$, $7$, $13$, $\ldots$, $2203$, $2281$, $3217$, $4253$, $4423$, $9689$, $9941$, $11 \, 213$, $19 \, 937$, $21 \, 701$:
::$M_{23 \, 209} = 2^{23 \, 209} - 1 \approx 4 \cdotp 029 \times 10^{6986}$
== Also see ==
* 
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== Historical Note ==
== Sources ==
* : $2^{23209} - 1$
* : $2^{23,209} - 1$
Category:Specific Numbers
Category:23,209
23,209/Historical Note

== Historical Note on $23 \, 209$ ==
The Mersenne number $M_{23 \, 209} = 2^{23 \, 209} - 1$ was demonstrated to be a Mersenne prime on $9$ February $1979$ by , using the same CDC-CYBER-174 that he and used to discover the primality of $M_{21 \, 701}$.
It took the computer $8$ hours to check the number.
$2$ weeks later, used a CRAY-1 supercomputer to check the result, and it took $7$ minutes.
== Sources ==
* : $2^{23209} - 1$
* : $2^{23,209} - 1$
Category:Historical Notes
23,456,789

== Number ==
$23 \, 456 \, 789$ is:
:The $1 \, 475 \, 171$st prime number
:The smallest prime number with all digits except $0$ and $1$
:The $11$th prime number after $2$, $3$, $5$, $7$, $23$, $67$, $89$, $4567$, $78 \, 901$, $678 \, 901$ consisting of a string of consecutive ascending digits (allowing $0$ to follow $9$)
:Also note that $23$, $4567$ and $89$ are all prime numbers also
== Also see ==
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== Sources ==
* : $23,456,789$
Category:Specific Numbers
Category:23,456,789
23,801

== Number ==
$23 \, 801$ (twenty-three thousand, eight hundred and one) is:
:The $2647$th prime number
:The $17$th prime $p$ such that $p \# + 1$, where $p \#$ denotes primorial (product of all primes up to $p$) of $p$, is prime:
::$2$, $3$, $5$, $7$, $11$, $31$, $379$, $1019$, $1021$, $2657$, $3229$, $4547$, $4787$, $11 \, 549$, $13 \, 649$, $18 \, 523$, $23 \, 801$
== Also see ==
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Category:Specific Numbers
Category:23,801
230

== Number ==
$230$ (two hundred and thirty) is:
:$2 \times 5 \times 23$
:The $21$st sphenic number after $30$, $42$, $66$, $70$, $78$, $102$, $105$, $110$, $114$, $130$, $138$, $154$, $165$, $170$, $174$, $182$, $186$, $190$, $195$, $222$:
::$230 = 2 \times 5 \times 23$
:The $33$rd nontotient:
::$\nexists m \in \Z_{>0}: \map \phi m = 230$
:where $\map \phi m$ denotes the Euler $\phi$ function
:The $37$th happy number after $1$, $7$, $10$, $13$, $19$, $23$, $\ldots$, $130$, $133$, $139$, $167$, $176$, $188$, $190$, $192$, $193$, $203$, $208$, $219$, $226$:
::$230 \to 2^2 + 3^2 + 0^2 = 4 + 9 + 0 = 13 \to 1^2 + 3^2 = 1 + 9 = 10 \to 1^2 + 0^2 = 1$
:There are $230$ Fedorov groups, if chiral copies are considered distinct.
== Also see ==
* 230 Fedorov Groups where Chiral Pairs are Distinct
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== Sources ==
* : $219$
* : $219$
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Category:Specific Numbers
Category:230
2300

== Number ==
$2300$ (two thousand three hundred) is:
:$2^2 \times 5^2 \times 23$
:The $23$rd tetrahedral number, after $1$, $4$, $10$, $20$, $35$, $\ldots$, $816$, $969$, $1140$, $1330$, $1540$, $1771$, $2024$:
::$2300 = \ds \sum_{k \mathop = 1}^{23} \frac {k \paren {k + 1} } 2 = \dfrac {23 \paren {23 + 1} \paren {23 + 2} } 6$
== Also see ==
* 
Category:Specific Numbers
Category:2300
2301

== Number ==
$2301$ (two thousand, three hundred and two) is:
:$3 \times 13 \times 59$
:The $39$th second pentagonal number after $2$, $7$, $15$, $26$, $\ldots$, $1107$, $1190$, $1276$, $1365$, $1457$, $1552$, $1650$, $1751$, $1855$, $1962$, $2072$, $2185$:
::$2301 = \dfrac {39 \paren {3 \times 39 + 1} } 2$
:The $78$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $1617$, $1650$, $1717$, $1751$, $1820$, $1855$, $1926$, $1962$, $2072$, $2147$, $2185$, $2262$:
::$2301 = \dfrac {39 \paren {3 \times 39 + 1} } 2$
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Category:Specific Numbers
Category:2301
2304

== Number ==
$2304$ (two thousand, three hundred and four) is:
:$2^8 \times 3^2$
:The $18$th square number after $1$, $4$, $36$, $121$, $144$, $256$, $324$, $400$, $576$, $784$, $900$, $961$, $1024$, $1296$, $1600$, $1764$, $1936$ to be the divisor sum value of some (strictly) positive integer:
::$2304 = \map {\sigma_1} {930}$
:The $42$nd positive integer which cannot be expressed as the sum of a square and a prime:
::$1$, $10$, $25$, $34$, $58$, $64$, $85$, $\ldots$, $1024$, $1089$, $1225$, $1255$, $1351$, $1414$, $1444$, $1521$, $1681$, $1849$, $1906$, $1936$, $2116$, $2209$, $2304$, $\ldots$
:The $48$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $1296$, $1369$, $1444$, $1521$, $1600$, $1681$, $1764$, $1849$, $1936$, $2025$, $2116$, $2209$:
::$2304 = 48 \times 48$
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Category:Specific Numbers
Category:2304
2309

== Number ==
$2309$ (two thousand, three hundred and-nine) is:
:The $343$rd prime number
:The $3$rd primorial prime after $5$, $29$:
::$2309 = p_5 \# - 1 = 11 \# - 1 = 2 \times 3 \times 5 \times 7 \times 11 - 1$
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Category:Specific Numbers
Category:2309
231

== Number ==
$231$ (two hundred and thirty-one) is:
:$3 \times 7 \times 11$
:The $11$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $120$, $153$, $190$:
::$231 = \ds \sum_{k \mathop = 1}^{11} \paren {4 k - 3} = 11 \paren {2 \times 11 - 1}$
:The number of integer partitions for $16$:
::$\map p {16} = 231$
:The $21$st triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $136$, $153$, $171$, $190$, $210$:
::$231 = \ds \sum_{k \mathop = 1}^{21} k = \dfrac {21 \times \paren {21 + 1} } 2$
:The $22$nd sphenic number after $30$, $42$, $66$, $70$, $78$, $102$, $105$, $110$, $114$, $130$, $138$, $154$, $165$, $170$, $174$, $182$, $186$, $190$, $195$, $222$, $230$:
::$231 = 3 \times 7 \times 11$
:The $44$th lucky number:
::$1$, $3$, $7$, $9$, $13$, $15$, $21$, $\ldots$, $163$, $169$, $171$, $189$, $193$, $195$, $201$, $205$, $211$, $219$, $223$, $231$, $\ldots$
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Category:Specific Numbers
Category:231
2310

== Number ==
$2310$ (two thousand, three hundred and ten) is:
:$2 \times 3 \times 5 \times 7 \times 11$
:The $5$th primorial after $1$, $2$, $6$, $30$, $210$ (counting $1$ as the zeroth):
::$2310 = p_5 \# = 11 \# = 2 \times 3 \times 5 \times 7 \times 11$
:Hence the smallest positive integer with $5$ distinct prime factors.
== Also see ==
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== Sources ==
* : $2310$
* : $2310$
Category:Specific Numbers
Category:2310
2311

== Number ==
$2311$ (two thousand, three hundred and eleven) is:
:The $344$th prime number
:The $5$th Euclid number after $3$, $7$, $31$, $211$:
::$2311 = p_5\# + 1 = 2 \times 3 \times 5 \times 7 \times 11 + 1$
:The $6$th Euclid prime after $2$, $3$, $7$, $31$, $211$:
::$2311 = p_5\# + 1 = 2 \times 3 \times 5 \times 7 \times 11 + 1$
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Category:Specific Numbers
Category:2311
232

== Number ==
$232$ (two hundred and thirty-two) is:
:$2^3 \times 29$
:The $22$nd noncototient after $10$, $26$, $34$, $50$, $52$, $58$, $\ldots$, $170$, $172$, $186$, $202$, $206$, $218$, $222$:
::$\nexists m \in \Z_{>0}: m - \map \phi m = 232$
:where $\map \phi m$ denotes the Euler $\phi$ function
:The $1$st element of the smallest triple of consecutive positive integers each of which is the sum of two squares:
::$232 = 14^2 + 6^2$, $233 = 13^2 + 8^2$, $234 = 15^2 + 3^2$
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== Sources ==
* : $232$
* : $232$
Category:Specific Numbers
Category:232