nyuuzyou/wikis huggingface dataset:
2147

== Number ==
$2147$ (two thousand, one hundred and forty-seven) is:
:$19 \times 113$
:The $38$th pentagonal number after $1$, $5$, $12$, $22$, $35$, $\ldots$, $925$, $1001$, $1080$, $1162$, $1247$, $1335$, $1426$, $1520$, $1617$, $1717$, $1820$, $1926$, $2035$:
::$2147 = \ds \sum_{k \mathop = 1}^{38} \paren {3 k - 2} = \dfrac {38 \paren {3 \times 38 - 1} } 2$
:The $75$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $1617$, $1650$, $1717$, $1751$, $1820$, $1855$, $1926$, $1962$, $2072$:
::$2147 = \ds \sum_{k \mathop = 1}^{38} \paren {3 k - 2} = \dfrac {38 \paren {3 \times 38 - 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:2147
215,326

== Number ==
$215\,326$ (two hundred and fifteen thousand, three hundred and twenty-six) is:
:$2 \times 23 \times 31 \times 151$
:The $2$nd even Poulet number after $161\,038$:
::$2^{215\,326} \equiv 2 \pmod {215\,326}$: $215\,326 = 2 \times 23 \times 31 \times 151$
== Also see ==
* 
== Sources ==
* : $161,038$
Category:Specific Numbers
Category:215,326
216

== Number ==
$216$ (two hundred and sixteen) is:
:$2^3 \times 3^3$
:The $1$st cube which can be expressed as the sum of $3$ positive cubes:
::$216 = 3^3 + 4^3 + 5^3$
:The $1$st 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 $3$rd power of $6$ after $(1)$, $6$, $36$:
::$216 = 6^3$
:The $6$th cube number after $1$, $8$, $27$, $64$, $125$:
::$216 = 6 \times 6 \times 6$
:The $7$th positive integer after $64$, $96$, $128$, $144$, $160$, $192$ with $6$ or more prime factors:
::$216 = 2 \times 2 \times 2 \times 3 \times 3 \times 3$
:The $8$th Friedman number base $10$ after $25$, $121$, $125$, $126$, $127$, $128$, $153$:
::$216 = 6^{1 + 2}$
:The $10$th inconsummate number after $62$, $63$, $65$, $75$, $84$, $95$, $161$, $173$, $195$:
::$\nexists n \in \Z_{>0}: n = 216 \times \map {s_{10} } n$
:The $12$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 $13$th untouchable number after $2$, $5$, $52$, $88$, $96$, $120$, $124$, $146$, $162$, $188$, $206$, $210$
:The $24$th Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$, $12$, $15$, $24$, $36$, $111$, $112$, $115$, $128$, $132$, $135$, $144$, $175$, $212$:
::$216 = 18 \times 12 = 18 \times \paren {2 \times 1 \times 6}$
:The $24$th powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $\ldots$, $108$, $121$, $125$, $128$, $144$, $169$, $196$, $200$
:The $28$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $12$, $16$, $18$, $20$, $24$, $30$, $36$, $42$, $48$, $60$, $72$, $84$, $90$, $96$, $108$, $120$, $144$, $168$, $180$, $210$:
::$\map {\sigma_1} {216} = 600$
:The $42$nd positive integer $n$ such that no factorial of an integer can end with $n$ zeroes.
== Also see ==
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
== Historical Note ==
== Sources ==
* : Glossary
* : $216$
* : Glossary
* : $216$
Category:Specific Numbers
Category:216
216,091

== Number ==
$216 \, 091$ (two hundred and sixteen thousand and ninety-one) is:
:The $19 \, 292$nd prime number
:The index of the $31$st Mersenne prime after $2$, $3$, $5$, $7$, $13$, $\ldots$, $11 \, 213$, $19 \, 937$, $21 \, 701$, $23 \, 209$, $44 \, 497$, $86 \, 243$, $110 \, 503$, $132 \, 049$:
::$M_{216 \, 091} = 2^{216 \, 091} - 1 \approx 7 \cdotp 461 \times 10^{65 \, 049}$
== Also see ==
* 
* 
== Historical Note ==
== Sources ==
* : $2^{216091} - 1$
Category:Specific Numbers
Category:216,091
216,091/Historical Note

== Historical Note on $216 \, 091$ ==
, in his of $1986$, announces that the Mersenne number $M_{261 \, 092} = 2^{261 \, 092} - 1$ was discovered by researchers at Chevron Geosciences of Houston, Texas, running performance tests on their Cray X-MP, in preparation for its use in oil prospecting.
It took $3$ hours, at $400$ million operations per second, to accomplish this feat, although the preliminary preparation to determine its possible primeness had taken several months beforehand.
An announcement about this was made on $18$ September $1985$, and, according to , was broadcast to Britain on BBC Radio.
However, it appears that had already discovered this fact on $1$ September $1985$, probably using the same (or a similar) configuration of the CRAY-1 he had used for previous searches.
Further research is needed into the exact details.
== Sources ==
* : $2^{216091} - 1$
Category:Historical Notes
2161

== Number ==
$2161$ (two thousand, one hundred and sixty-one) is:
:The $326$th prime number
:The $3$rd integer $m$ after $1$, $47$ whose cube can be expressed as the sum of $m$ consecutive squares:
::$2161^3 = \ds \sum_{k \mathop = 1}^{2161} \paren {989 + k}^2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:2161
217

== Number ==
$217$ (two hundred and seventeen) is:
:$7 \times 31$
:The $3$rd Fermat pseudoprime to base $5$ after $4$, $124$:
::$5^{217} \equiv 5 \pmod {217}$
:The $9$th centered hexagonal number after $1$, $7$, $19$, $37$, $61$, $91$, $127$, $169$:
::$217 = 1 + 6 + 12 + 18 + 24 + 30 + 36 + 42 + 48 = 9^3 - 8^3$
:The $13$th number after $1$, $3$, $22$, $66$, $70$, $81$, $94$, $115$, $119$, $170$, $210$, $214$ whose divisor sum is square:
:
:The $43$rd positive integer $n$ such that no factorial of an integer can end with $n$ zeroes.
== Also see ==
* 
* 
* 
* 
== Sources ==
* : $217$
Category:Specific Numbers
Category:217
2176

== Number ==
$2176$ (two thousand, one hundred and seventy-six) is:
:$2^7 \times 17$
:The $16$th pentagonal pyramidal number after $1$, $6$, $12$, $40$, $75$, $126$, $196$, $288$, $405$, $550$, $726$, $936$, $1183$, $1470$, $1800$:
::$2176 = \ds \sum_{k \mathop = 1}^{16} \dfrac {k \paren {3 k - 1} } 2 = \dfrac {16^2 \paren {16 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:2176
2178

== Number ==
$2178$ (two thousand, one hundred and seventy-eight) is:
:$2 \times 3^2 \times 11^2$
:The $2$nd of only $2$ numbers with $4$ digits or fewer which has a multiple which is its reversal:
::$2178 \times 4 = 8712$
:A $4$th order recurring digital invariant:
::$2^4 + 1^4 + 7^4 + 8^4 = 6514$; $6^4 + 5^4 + 1^4 + 4^4 = 2178$
== Also see ==
* 
== Sources ==
* : $1089$
* : $2178$
* : $1089$
* : $2178$
Category:Specific Numbers
Category:2178
218

== Number ==
$218$ (two hundred and eighteen) is:
:$2 \times 109$
:The $32$nd nontotient:
::$\nexists m \in \Z_{>0}: \phi \left({m}\right) = 218$
:where $\phi \left({m}\right)$ denotes the Euler $\phi$ function
:The $20$th noncototient after $10$, $26$, $34$, $50$, $52$, $58$, $86$, $100$, $116$, $122$, $130$, $134$, $146$, $154$, $170$, $172$, $186$, $202$, $206$:
::$\nexists m \in \Z_{>0}: m - \phi \left({m}\right) = 218$
:where $\phi \left({m}\right)$ denotes the Euler $\phi$ function
== Also see ==
* 
* 
Category:Specific Numbers
Category:218
2184

== Number ==
$2184$ (two thousand, one hundred and eighty-four) is:
:$2^3 \times 3 \times 7 \times 13$
:The $1$st term of a sequence of $17$ consecutive integers all of which are not coprime with the product of the other $16$:
::$2184 \not \perp \ds \prod_{k \mathop = 2184}^{2200} k$
== Also see ==
* 
== Sources ==
* : $2185$
Category:Specific Numbers
Category:2184
2185

== Number ==
$2185$ (two thousand, one hundred and eighty-five) is:
:$5 \times 19 \times 23$
:The $2$nd term of a sequence of $17$ consecutive integers all of which are not coprime with the product of the other $16$:
::$2185 \not \perp \ds \prod_{\substack {2184 \mathop \le k \mathop \le 2200 \\ k \mathop \ne 2185} } k$
:The $38$th second pentagonal number after $2$, $7$, $15$, $26$, $\ldots$, $1107$, $1190$, $1276$, $1365$, $1457$, $1552$, $1650$, $1751$, $1855$, $1962$, $2072$:
::$2185 = \dfrac {38 \paren {3 \times 38 + 1} } 2$
:The $76$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $1617$, $1650$, $1717$, $1751$, $1820$, $1855$, $1926$, $1962$, $2072$, $2147$:
::$2185 = \dfrac {38 \paren {3 \times 38 + 1} } 2$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:2185
2186

== Number ==
$2186$ (two thousand, one hundred and eighty-six) is:
:$2 \times 1093$
:The $3$rd term of a sequence of $17$ consecutive integers all of which are not coprime with the product of the other $16$:
::$2186 \not \perp \ds \prod_{\substack {2184 \mathop \le k \mathop \le 2200 \\ k \mathop \ne 2186} } k$
== Also see ==
* 
Category:Specific Numbers
Category:2186
2187

== Number ==
$2187$ (two thousand, one hundred and eighty-seven) is:
:$3^7$
:The $3$rd seventh power after $1$, $128$:
::$2187 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3$
:Thus in ternary:
::$2187_{10} = 10 \, 000 \, 000_3$
:The $4$th term of a sequence of $17$ consecutive integers all of which are not coprime with the product of the other $16$:
::$2187 \mathrel \not \perp \ds \prod_{\substack {2184 \mathop \le k \mathop \le 2200 \\ k \mathop \ne 2187} } k$
:The $7$th power of $3$ after $(1)$, $3$, $9$, $27$, $81$, $243$, $729$:
::$2187 = 3^7$
== Also see ==
* 
* 
* 
== Sources ==
* : $2187$
* : $2187$
Category:Specific Numbers
Category:2187
2188

== Number ==
$2188$ (two thousand, one hundred and eighty-eight) is:
:$2^2 \times 547$
:The $5$th term of a sequence of $17$ consecutive integers all of which are not coprime with the product of the other $16$:
::$2188 \not \perp \ds \prod_{\substack {2184 \mathop \le k \mathop \le 2200 \\ k \mathop \ne 2188} } k$
== Also see ==
* 
Category:Specific Numbers
Category:2188
2189

== Number ==
$2189$ (two thousand, one hundred and eighty-nine) is:
:$11 \times 199$
:The $6$th term of a sequence of $17$ consecutive integers all of which are not coprime with the product of the other $16$:
::$2189 \not \perp \ds \prod_{\substack {2184 \mathop \le k \mathop \le 2200 \\ k \mathop \ne 2189} } k$
== Also see ==
* 
Category:Specific Numbers
Category:2189
219

== Number ==
$219$ (two hundred and nineteen) is:
:$3 \times 73$
:The $42$nd lucky number:
::$1$, $3$, $7$, $9$, $13$, $15$, $21$, $\ldots$, $163$, $169$, $171$, $189$, $193$, $195$, $201$, $205$, $211$, $219$, $\ldots$
:The $35$th happy number after $1$, $7$, $10$, $13$, $19$, $23$, $\ldots$, $130$, $133$, $139$, $167$, $176$, $188$, $190$, $192$, $193$, $203$, $208$:
::$219 \to 2^2 + 1^2 + 9^2 = 4 + 1 + 81 = 86 \to 8^2 + 6^2 = 64 + 36 = 100 \to 1^2 + 0^2 + 0^2 = 1$
:The $2$nd of $4$ integers $n$ such that $n + 2 n$ can be expressed as a sum using each of the digits $1$ to $9$ exactly once each:
::$219 + 438 = 657$
:There are $219$ Fedorov groups.
== Also see ==
* 219 Fedorov Groups
* 
* 
* 
== Sources ==
* : $219$
* : $192$
* : $219$
Category:Specific Numbers
Category:219
2190

== Number ==
$2190$ (two thousand, one hundred and ninety) is:
:$2 \times 3 \times 5 \times 73$
:The $7$th term of a sequence of $17$ consecutive integers all of which are not coprime with the product of the other $16$:
::$2190 \not \perp \ds \prod_{\substack {2184 \mathop \le k \mathop \le 2200 \\ k \mathop \ne 2190} } k$
== Also see ==
* 
Category:Specific Numbers
Category:2190
2191

== Number ==
$2191$ (two thousand, one hundred and ninety-one) is:
:$7 \times 313$
:The $8$th term of a sequence of $17$ consecutive integers all of which are not coprime with the product of the other $16$:
::$2191 \not \perp \ds \prod_{\substack {2184 \mathop \le k \mathop \le 2200 \\ k \mathop \ne 2191} } k$
== Also see ==
* 
Category:Specific Numbers
Category:2191
2192

== Number ==
$2192$ (two thousand, one hundred and ninety-two) is:
:$2^4 \times 137$
:The $9$th term of a sequence of $17$ consecutive integers all of which are not coprime with the product of the other $16$:
::$2192 \not \perp \ds \prod_{\substack {2184 \mathop \le k \mathop \le 2200 \\ k \mathop \ne 2192} } k$
== Also see ==
* 
Category:Specific Numbers
Category:2192
2193

== Number ==
$2193$ (two thousand, one hundred and ninety-three) is:
:$3 \times 17 \times 43$
:The $10$th term of a sequence of $17$ consecutive integers all of which are not coprime with the product of the other $16$:
::$2193 \not \perp \ds \prod_{\substack {2184 \mathop \le k \mathop \le 2200 \\ k \mathop \ne 2193} } k$
== Also see ==
* 
Category:Specific Numbers
Category:2193
2194

== Number ==
$2194$ (two thousand, one hundred and ninety-four) is:
:$2 \times 1097$
:The $11$th term of a sequence of $17$ consecutive integers all of which are not coprime with the product of the other $16$:
::$2194 \not \perp \ds \prod_{\substack {2184 \mathop \le k \mathop \le 2200 \\ k \mathop \ne 2194} } k$
== Also see ==
* 
Category:Specific Numbers
Category:2194
2195

== Number ==
$2195$ (two thousand, one hundred and ninety-five) is:
:$5 \times 439$
:The $12$th term of a sequence of $17$ consecutive integers all of which are not coprime with the product of the other $16$:
::$2195 \not \perp \ds \prod_{\substack {2184 \mathop \le k \mathop \le 2200 \\ k \mathop \ne 2195} } k$
== Also see ==
* 
Category:Specific Numbers
Category:2195
2196

== Number ==
$2196$ (two thousand, one hundred and ninety-six) is:
:$2^2 \times 3^2 \times 61$
:The $13$th term of a sequence of $17$ consecutive integers all of which are not coprime with the product of the other $16$:
::$2196 \not \perp \ds \prod_{\substack {2184 \mathop \le k \mathop \le 2200 \\ k \mathop \ne 2196} } k$
== Also see ==
* 
Category:Specific Numbers
Category:2196
2197

== Number ==
$2197$ (two thousand, one hundred and ninety-seven) is:
:$13^3$
:The $3$rd power of $13$ after $(1)$, $13$, $169$:
::$2197 = 13^3$
:The $13$th cube number after $1$, $8$, $27$, $64$, $125$, $216$, $343$, $512$, $729$, $1000$, $1331$, $1728$:
::$2197 = 13 \times 13 \times 13$
:The $14$th term of a sequence of $17$ consecutive integers all of which are not coprime with the product of the other $16$:
::$2197 \not \perp \ds \prod_{\substack {2184 \mathop \le k \mathop \le 2200 \\ k \mathop \ne 2197} } k$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:2197
2197 is Cube of 13

== Theorem ==
:$13^3 = 2197$
== Proof ==
By long multiplication:
 13
 x 13
-----
 39
 130
-----
 169
-----
then:
 x 169
 13
------
 507
 1690
-----
 2197
------
== Sources ==
* : Notes on the Exercises: Exercise $3$
Category:Numbers
2198

== Number ==
$2198$ (two thousand, one hundred and ninety-eight) is:
:$2 \times 7 \times 157$
:The $15$th term of a sequence of $17$ consecutive integers all of which are not coprime with the product of the other $16$:
::$2198 \not \perp \ds \prod_{\substack {2184 \mathop \le k \mathop \le 2200 \\ k \mathop \ne 2198} } k$
== Also see ==
* 
Category:Specific Numbers
Category:2198
2199

== Number ==
$2199$ (two thousand, one hundred and ninety-nine) is:
:$3 \times 733$
:The $16$th term of a sequence of $17$ consecutive integers all of which are not coprime with the product of the other $16$:
::$2199 \not \perp \ds \prod_{\substack {2184 \mathop \le k \mathop \le 2200 \\ k \mathop \ne 2199} } k$
== Also see ==
* 
Category:Specific Numbers
Category:2199
22

== Number ==
$22$ (twenty-two) is:
:$2 \times 11$
:The $2$nd integer after $1$ which equals the number of digits in its factorial:
::$22! = 1 \, 124 \, 000 \, 727 \, 777 \, 607 \, 680 \, 000$
:which has $22$ digits
:The $2$nd Smith number after $4$:
::$2 + 2 = 2 + 1 + 1 = 4$
:The smallest positive integer which can be expressed as the sum of $2$ odd primes in $3$ ways:
::$22 = 19 + 3 = 17 + 5 = 11 + 11$
:The $3$rd pentagonal number after $1$, $5$ which is also palindromic:
::$22 = 1 + 4 + 7 + 10 = \dfrac {4 \paren {3 \times 4 - 1}} 2$
:The $3$rd hexagonal pyramidal number after $1$, $7$:
::$22 = 1 + 6 + 15$
:The $3$rd number after $1$, $3$ whose divisor sum is square:
:
:The $4$th pentagonal number after $1$, $5$, $12$:
::$22 = 1 + 4 + 7 + 10 = \dfrac {4 \paren {3 \times 4 - 1}} 2$
:The $6$th palindromic integer after $0$, $1$, $2$, $3$, $11$ whose square is also palindromic integer
::$22^2 = 484$
:The $7$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$:
::$22 = \dfrac {4 \paren {3 \times 4 - 1}} 2$
:The $8$th semiprime after $4$, $6$, $9$, $10$, $14$, $15$, $21$:
::$22 = 2 \times 11$
:The $8$th of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:
::$3$, $4$, $9$, $10$, $12$, $16$, $17$, $22$, $\ldots$
:The number of integer partitions for $8$:
::$\map p 8 = 22$
:The $10$th even number after $2$, $4$, $6$, $8$, $10$, $12$, $14$, $16$, $20$ which cannot be expressed as the sum of $2$ composite odd numbers.
:The $11$th positive integer which is not the sum of $1$ or more distinct squares:
::$2$, $3$, $6$, $7$, $8$, $11$, $12$, $15$, $18$, $19$, $22$, $\ldots$
== Also see ==
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
== Sources ==
* : $22$
* : $22$
Category:Specific Numbers
Category:22
22,140

== Number ==
$22 \, 140$ (twenty-two thousand, one hundred and forty) is:
:$2^2 \times 3^3 \times 5 \times 41$
:The $4$th square pyramorphic number after $1$, $55$, $5525$:
::$22 \, 140 = \ds \sum_{k \mathop = 1}^{40} k^2 = \dfrac {40 \paren {40 + 1} \paren {2 \times 40 + 1} } 6$
:The $40$th square pyramidal number:
::$22 \, 140 = \ds \sum_{k \mathop = 1}^{40} k^2 = \dfrac {40 \paren {40 + 1} \paren {2 \times 40 + 1} } 6$
== Also see ==
* 
* 
Category:Specific Numbers
Category:22,140
22,222,222,222,222

== Number ==
$22 \, 222 \, 222 \, 222 \, 222$ is:
:$2 \times 11 \times 239 \times 4649 \times 909 \, 091$
:The $284$th Kaprekar number:
::$22 \, 222 \, 222 \, 222 \, 222^2= 493 \, 827 \, 160 \, 493 \, 817 \, 283 \, 950 \, 617 \, 284 \to 4 \, 938 \, 271 \, 604 \, 938 + 17 \, 283 \, 950 \, 617 \, 284 = 22 \, 222 \, 222 \, 222 \, 222$
== Also see ==
* 
== Sources ==
* : $22,222,222,222,222$
* : $22,222,222,222,222$
Category:Specific Numbers
Category:22,222,222,222,222
22,440

== Number ==
$22 \, 440$ (twenty-two thousand, four hundred and forty) is:
:$2^3 \times 3 \times 5 \times 11 \times 17$
:The $2$nd positive integer $n$ after $15$ such that $\pm n$ allows $x^5 - x + n$ to be factorized into the product of an irreducible quadratic and an an irreducible cubic:
:$x^5 - x \pm 22440 = \paren {x^2 \mp 12 x + 55} \paren {x^3 \pm 12 x^2 + 89 x \pm 408}$
== Also see ==
* 
Category:Specific Numbers
Category:22,440
22,624

== Number ==
$22 \, 624$ (twenty-two thousand, six hundred and twenty-four) is:
:$2^5 \times 7 \times 101$
:The $1$st term of the $1$st quadruplet of consecutive positive integers each of which is divisible by a cube greater than $1$:
::$22 \, 624 = 2828 \times 2^3$, $22 \, 625 = 181 \times 5^3$, $22 \, 626 = 838 \times 3^3$, $22 \, 627 = 17 \times 11^3$
== Also see ==
* 
== Sources ==
* : $1375$
Category:Specific Numbers
Category:22,624
22,625

== Number ==
$22 \, 625$ (twenty-two thousand, six hundred and twenty-five) is:
:$5^3 \times 181$
:The $2$nd term of the $1$st quadruplet of consecutive positive integers each of which is divisible by a cube greater than $1$:
::$22 \, 624 = 2828 \times 2^3$, $22 \, 625 = 181 \times 5^3$, $22 \, 626 = 838 \times 3^3$, $22 \, 627 = 17 \times 11^3$
== Also see ==
* 
Category:Specific Numbers
Category:22,625
22,626

== Number ==
$22 \, 626$ (twenty-two thousand, six hundred and twenty-six) is:
:$2 \times 3^3 \times 419$
:The $3$rd term of the $1$st quadruplet of consecutive positive integers each of which is divisible by a cube greater than $1$:
::$22 \, 624 = 2828 \times 2^3$, $22 \, 625 = 181 \times 5^3$, $22 \, 626 = 838 \times 3^3$, $22 \, 627 = 17 \times 11^3$
== Also see ==
* 
Category:Specific Numbers
Category:22,626
22,627

== Number ==
$22 \, 627$ (twenty-two thousand, six hundred and twenty-seven) is:
:$11^3 \times 17$
:The $3$rd term of the $1$st quadruplet of consecutive positive integers each of which is divisible by a cube greater than $1$:
::$22 \, 624 = 2828 \times 2^3$, $22 \, 625 = 181 \times 5^3$, $22 \, 626 = 838 \times 3^3$, $22 \, 627 = 17 \times 11^3$
== Also see ==
* 
Category:Specific Numbers
Category:22,627
22,744

== Number ==
$22 \, 744$ (twenty-two thousand, seven hundred and forty-four) is:
:$2^3 \times 2843$
:The $26$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:22,744
22,976

== Number ==
$22 \, 976$ (twenty-two thousand, nine hundred and seventy-six) is:
:$2^6 \times 359$
:The $25$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:22,976
220

== Number ==
$220$ (two hundred and twenty) is:
:$2^2 \times 5 \times 11$
:The smaller of the $1$st amicable pair, with $284$:
::$\map {\sigma_1} {220} = \map {\sigma_1} {284} = 504 = 220 + 284$
:which is also the $1$st Thabit pair:
::$220 = 2^2 \times 11 \times 5 = 2^2 \paren {3 \times 2^2 - 1} \paren {3 \times 2^{2 - 1} - 1}$, $284 = 2^2 \paren {9 \times 2^{2 \times 2 - 1} - 1}$
:The $10$th tetrahedral number, after $1$, $4$, $10$, $20$, $35$, $56$, $84$, $120$, $165$:
::$220 = 1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45 + 55 = \dfrac {10 \paren {10 + 1} \paren {10 + 2} } 6$
== Also see ==
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== Historical Note ==
== Sources ==
* : $220$
* : $220$
Category:Specific Numbers
Category:220
220/Historical Note

== Historical Note on $220$ ==
The numbers $220$ and $284$, being the smallest amicable pair, were traditionally used in conjunction with overtures of friendship. 
The number $220$ appears in the Bible as the number of goats that Jacob gave to Esau on the event of their reunion.
It suggests that they may have been known about considerably earlier than the time of .
== Sources ==
* : $220$
* : $220$
Category:Historical Notes
2200

== Number ==
$2200$ (two thousand, two hundred) is:
:$2^3 \times 5^2 \times 11$
:The $17$th and final term of a sequence of $17$ consecutive integers all of which are not coprime with the product of the other $16$:
::$2200 \not \perp \ds \prod_{k \mathop = 2184}^{2199} k$
== Also see ==
* 
Category:Specific Numbers
Category:2200
2201

== Number ==
$2201$ (two thousand, two hundred and one) is:
:$31 \times 71$
:The $1$st non-palindromic positive integer (and the only one known) whose cube is palindromic:
::$2201^3 = 10 \, 662 \, 526 \, 601$
:The $8$th positive integer after $1$, $2$, $7$, $11$, $101$, $111$, $1001$ whose cube is palindromic:
::$2201^3 = 10 \, 662 \, 526 \, 601$
== Also see ==
* 
== Sources ==
* : $2201$
Category:Specific Numbers
Category:2201
2203

== Number ==
$2203$ (two thousand, two hundred and three) is:
:The $328$th prime number
:The index of the $16$th Mersenne prime after $2$, $3$, $5$, $7$, $13$, $17$, $19$, $31$, $61$, $89$, $107$, $127$, $521$, $607$, $1279$:
::$M_{2203} = 2^{2203} - 1 \approx 1 \cdotp 476 \times 10^{663}$
== Also see ==
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Category:Specific Numbers
Category:2203
2205

== Number ==
$2205$ (two thousand, two hundred and five) is:
:$3^2 \times 5 \times 7^2$
:The $30$th heptagonal number after $1$, $7$, $18$, $34$, $55$, $81$, $112$, $\ldots$, $1525$, $1651$, $1782$, $1918$, $2059$:
::$2205 = \ds \sum_{k \mathop = 1}^{30} \paren {5 k - 4} = \dfrac {30 \paren {5 \times 30 - 3} } 2$
:The $3$rd odd abundant number after $945$, $1575$:
::$\map {\sigma_1} {2205} - 2205 = 2241 > 2205$
== Also see ==
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Category:Specific Numbers
Category:2205