nyuuzyou/wikis huggingface dataset:
32,761

== Number ==
$32 \, 761$ (thirty-two thousand, seven hundred and sixty-one) is:
:$181^2$
:The $3$rd centered hexagonal number after $1, 169$ which is also square:
::$32 \, 761 = 105^3 - 104^3 = 181^2$
:The $105$th centered hexagonal number:
::$32 \, 761 = 3 \times 105 \times \paren {105 - 1} + 1 = 105^3 - 104^3$
:The $181$st square number:
::$32 \, 761 = 181 \times 181$
== Also see ==
* 
* 
* 
== Sources ==
* : $169$
Category:Specific Numbers
Category:32,761
32,768

== Number ==
$32 \, 768$ (thirty-two thousand, seven hundred and sixty-eight) is:
:$2^{15}$
:The $5$th power of $8$ after $(1)$, $8$, $64$, $512$, $4096$:
::$32 \, 768 = 8^5$
:The $8$th fifth power after $1$, $32$, $243$, $1024$, $3125$, $7776$, $16 \, 807$:
::$32 \, 768 = 8 \times 8 \times 8 \times 8 \times 8$
:The $15$th power of $2$ after $(1)$, $2$, $4$, $8$, $16$, $32$, $64$, $128$, $256$, $512$, $1024$, $2048$, $4096$, $8192$, $16 \, 384$:
::$32 \, 768 = 2^{15}$
:The $16$th almost perfect number after $1$, $2$, $4$, $8$, $16$, $32$, $64$, $128$, $256$, $512$, $1024$, $2048$, $4096$, $8192$, $16 \, 384$:
::$\map {\sigma_1} {32 \, 768} = 65 \, 535 = 2 \times 32 \, 768 - 1$
:The $32$nd cube number:
::$32 \, 768 = 32 \times 32 \times 32$
== Also see ==
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:32,768
320

== Number ==
$320$ (three hundred and twenty) is:
:$2^6 \times 5$
:The $5$th positive integer after $128$, $192$, $256$, $288$ with $7$ or more prime factors:
::$320 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 5$
:The $7$th positive integer after $200$, $202$, $204$, $205$, $206$, $208$ that cannot be made into a prime number by changing just $1$ digit
:The $12$th positive integer after $64$, $96$, $128$, $144$, $160$, $192$, $216$, $224$, $240$, $256$, $288$ with $6$ or more prime factors:
::$320 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \paren {\times \, 5}$
:The $13$th integer $m$ such that $m! + 1$ (its factorial plus $1$) is prime:
::$0$, $1$, $2$, $3$, $11$, $27$, $37$, $41$, $73$, $77$, $116$, $154$, $320$
:The $50$th happy number after $1$, $7$, $10$, $13$, $19$, $23$, $\ldots$, $226$, $230$, $236$, $239$, $262$, $263$, $280$, $291$, $293$, $301$, $302$, $310$, $313$, $319$:
::$320 \to 3^2 + 2^2 + 0^2 = 9 + 4 + 0 = 13 \to 1^2 + 3^2 = 1 + 9 = 10 \to 1^2 + 0^2 = 1$
== Also see ==
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:320
3201

== Number ==
$3201$ (three thousand, two hundred and one) is:
:$3 \times 11 \times 97$
:The $33$rd octagonal number, after $1$, $8$, $21$, $40$, $65$, $\ldots$, $1281$, $1408$, $1541$, $1680$, $1825$, $1976$, $2133$, $2296$, $2465$, $2640$, $2821$, $3008$:
::$3201 = \ds \sum_{k \mathop = 1}^{33} \paren {6 k - 5} = 33 \paren {3 \times 33 - 2}$
== Also see ==
* 
Category:Specific Numbers
Category:3201
3217

== Number ==
$3217$ (three thousand, two hundred and seventeen) is:
:The $455$th prime number
:The index of the $18$th Mersenne prime after $2$, $3$, $5$, $7$, $13$, $17$, $19$, $31$, $61$, $89$, $107$, $127$, $521$, $607$, $1279$, $2203$, $2281$:
::$M_{3217} = 2^{3217} - 1 \approx 2 \cdotp 591 \times 10^{968}$
== Also see ==
* 
* 
Category:Specific Numbers
Category:3217
322

== Number ==
$322$ (three hundred and twenty-two) is:
:$2 \times 7 \times 23$
:The $8$th positive integer after $200$, $202$, $204$, $205$, $206$, $208$, $320$ that cannot be made into a prime number by changing just $1$ digit
:The $12$th Lucas number after $(2)$, $1$, $3$, $4$, $7$, $11$, $18$, $29$, $47$, $76$, $123$, $199$:
::$322 = 123 + 199$
:The $17$th number after $1$, $3$, $22$, $66$, $70$, $81$, $94$, $115$, $119$, $170$, $210$, $214$, $217$, $265$, $282$, $310$ whose divisor dum is square:
:
:The $25$th untouchable number after $2$, $5$, $52$, $88$, $96$, $120$, $124$, $\ldots$, $262$, $268$, $276$, $288$, $290$, $292$, $304$, $306$
:The $35$th sphenic number after $30$, $42$, $66$, $70$, $\ldots$, $246$, $255$, $258$, $266$, $273$, $282$, $285$, $286$, $290$, $310$, $318$:
::$322 = 2 \times 7 \times 23$
== Also see ==
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:322
3229

== Number ==
$3229$ (three thousand, two hundred and twenty-nine) is:
:The $457$th prime number
:The $11$th prime $p$ such that $p \# + 1$, where $p \#$ denotes primorial (product of all primes up to $p$) of $p$, is prime:
::$2$, $3$, $5$, $7$, $11$, $31$, $379$, $1019$, $1021$, $2657$, $3229$
== Also see ==
* 
* 
Category:Specific Numbers
Category:3229
323

== Number ==
$323$ (three hundred and twenty-three) is:
:$17 \times 19$
:The $3$rd term of the smallest triplet of integers in arithmetic sequence which have the same divisor sum:
::$\map {\sigma_1} {267} = \map {\sigma_1} {295} = \map {\sigma_1} {323} = 360$
== Also see ==
* 
== Sources ==
* : $267$
Category:Specific Numbers
Category:323
324

== Number ==
$324$ (three hundred and twenty-four) is:
:$2^2 \times 3^4$
:The $7$th square number after $1$, $4$, $36$, $121$, $144$, $256$ to be the divisor sum value of some (strictly) positive integer:
::$324 = \map {\sigma_1} {170} = \map {\sigma_1} {214} = \map {\sigma_1} {265}$
:The $9$th positive integer after $200$, $202$, $204$, $205$, $206$, $208$, $320$, $322$ that cannot be made into a prime number by changing just $1$ digit
:The $13$th integer $m$ such that $m! - 1$ (its factorial minus $1$) is prime:
::$3$, $4$, $6$, $7$, $12$, $14$, $30$, $32$, $33$, $38$, $94$, $166$, $324$
:The $13$th positive integer after $64$, $96$, $128$, $144$, $160$, $192$, $216$, $224$, $240$, $256$, $288$, $320$ with $6$ or more prime factors:
::$324 = 2 \times 2 \times 3 \times 3 \times 3 \times 3$
:The $16$th positive integer which cannot be expressed as the sum of a square and a prime:
::$1$, $10$, $25$, $34$, $58$, $64$, $85$, $91$, $121$, $130$, $169$, $196$, $214$, $226$, $289$, $324$, $\ldots$
:The $18$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $49$, $64$, $81$, $100$, $121$, $144$, $169$, $196$, $225$, $256$, $289$:
::$324 = 18 \times 18$
:The $26$th untouchable number after $2$, $5$, $52$, $88$, $96$, $120$, $124$, $\ldots$, $262$, $268$, $276$, $288$, $290$, $292$, $304$, $306$, $322$
:The $30$th powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $\ldots$, $144$, $169$, $196$, $200$, $216$, $225$, $243$, $256$, $288$, $289$
== Also see ==
* 
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Category:Specific Numbers
Category:324
3240

== Number ==
$3240$ (three thousand, two hundred and forty) is:
:$2^3 \times 3^4 \times 5$
:The $80$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $2701$, $2775$, $2850$, $2926$, $3003$, $3081$, $3160$:
::$3240 = \ds \sum_{k \mathop = 1}^{80} k = \dfrac {80 \times \paren {80 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:3240
3244

== Number ==
$3244$ (three thousand, two hundred and forty-four) is:
:$2^2 \times 811$
:The $30$th even integer after $2$, $4$, $94$, $96$, $\ldots$, $1148$, $1264$, $1266$, $1268$, $1354$, $1356$, $1358$ that cannot be expressed as the sum of $2$ prime numbers which are each one of a pair of twin primes
== Also see ==
* 
Category:Specific Numbers
Category:3244
3246

== Number ==
$3246$ (three thousand, two hundred and forty-six) is:
:$2 \times 3 \times 541$
:The $31$st even integer after $2$, $4$, $94$, $96$, $\ldots$, $1148$, $1264$, $1266$, $1268$, $1354$, $1356$, $1358$, $3244$ that cannot be expressed as the sum of $2$ prime numbers which are each one of a pair of twin primes
== Also see ==
* 
Category:Specific Numbers
Category:3246
3248

== Number ==
$3248$ (three thousand, two hundred and forty-eight) is:
:$2^4 \times 7 \times 29$
:The $32$nd even integer after $2$, $4$, $94$, $96$, $\ldots$, $1148$, $1264$, $1266$, $1268$, $1354$, $1356$, $1358$, $3244$, $3246$ that cannot be expressed as the sum of $2$ prime numbers which are each one of a pair of twin primes
== Also see ==
* 
Category:Specific Numbers
Category:3248
3249

== Number ==
$3249$ (three thousand, two hundred and forty-nine) is:
:$3^2 \times 19^2$
:The $7$th square after $49$, $169$, $361$, $1225$, $1444$, $1681$ whose decimal representation can be split into two parts which are each themselves square:
::$3249 = 57^2$; $324 = 18^2$, $9 = 3^2$
:The $57$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $1936$, $2025$, $2116$, $2209$, $2304$, $2401$, $2500$, $2601$, $2704$, $2809$, $2916$, $3025$, $3136$:
::$3249 = 57 \times 57$
== Also see ==
* 
* 
Category:Specific Numbers
Category:3249
325

== Number ==
$325$ (three hundred and twenty-five) is:
:$5^2 \times 13$
:The $1$st positive integer which can be expressed as the sum of two square numbers in $3$ distinct ways:
::$325 = 18^2 + 1^2 = 17^2 + 6^2 = 15^2 + 10^2$
:The total of all the entries in a magic square of order $5$, after $1$, $(10)$, $45$, $136$:
::$325 = \ds \sum_{k \mathop = 1}^{5^2} k = \dfrac {5^2 \paren {5^2 + 1} } 2$
:The total number of permutations of $r$ objects from a set of $5$ objects, where $1 \le r \le 5$
:The $10$th positive integer after $200$, $202$, $204$, $205$, $206$, $208$, $320$, $322$, $324$ that cannot be made into a prime number by changing just $1$ digit
:The $13$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $120$, $153$, $190$, $231$, $276$:
::$325 = \ds \sum_{k \mathop = 1}^{13} \paren {4 k - 3} = 13 \paren {2 \times 13 - 1}$
:The $25$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $171$, $190$, $210$, $231$, $253$, $276$, $300$:
::$325 = \ds \sum_{k \mathop = 1}^{25} k = \dfrac {25 \times \paren {25 + 1} } 2$
== Also see ==
* 
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* 
* 
* 
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== Sources ==
* : $325$
* : $325$
Category:Specific Numbers
Category:325
326

== Number ==
$326$ (three hundred and twenty-six) is:
:$2 \times 173$
:The $11$th positive integer after $200$, $202$, $204$, $205$, $206$, $208$, $320$, $322$, $324$, $325$ that cannot be made into a prime number by changing just $1$ digit
:The $15$th inconsummate number after $62$, $63$, $65$, $75$, $84$, $95$, $161$, $173$, $195$, $216$, $261$, $266$, $272$, $276$:
::$\nexists n \in \Z_{>0}: n = 326 \times \map {s_{10} } n$
:The $27$th untouchable number after $2$, $5$, $52$, $88$, $96$, $120$, $124$, $\ldots$, $262$, $268$, $276$, $288$, $290$, $292$, $304$, $306$, $322$, $324$
:The $32$nd noncototient after $10$, $26$, $34$, $50$, $\ldots$, $244$, $260$, $266$, $268$, $274$, $290$, $292$, $298$, $310$:
::$\nexists m \in \Z_{>0}: m - \map \phi m = 326$
:where $\map \phi m$ denotes the Euler $\phi$ function
:The $51$st happy number after $1$, $7$, $10$, $13$, $19$, $23$, $\ldots$, $226$, $230$, $236$, $239$, $262$, $263$, $280$, $291$, $293$, $301$, $302$, $310$, $313$, $319$, $320$:
::$326 \to 3^2 + 2^2 + 6^2 = 9 + 4 + 36 = 49 \to 4^2 + 9^2 = 16 + 81 = 97 \to 9^2 + 7^2 = 81 + 49 = 130 \to 1^2 + 3^2 + 0^2 = 1 + 9 + 0 = 10 \to 1^2 + 0^2 = 1$
== Also see ==
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:326
327

== Number ==
$327$ (three hundred and twenty-seven) is:
:$3 \times 7 \times 13$
:The $4$th 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:
::$327 + 654 = 981$
== Also see ==
* 
== Sources ==
* : $192$
Category:Specific Numbers
Category:327
3276

== Number ==
$3276$ (three thousand, two hundred and seventy-six) is:
:$2^2 \times 3^2 \times 7 \times 13$
:The $26$th tetrahedral number, after $1$, $4$, $10$, $20$, $35$, $\ldots$, $816$, $969$, $1140$, $1330$, $1540$, $1771$, $2024$, $2300$, $2600$, $2925$:
::$3276 = \ds \sum_{k \mathop = 1}^{26} \frac {k \paren {k + 1} } 2 = \dfrac {26 \paren {26 + 1} \paren {26 + 2} } 6$
== Also see ==
* 
Category:Specific Numbers
Category:3276
3277

== Number ==
$3277$ (three thousand, two hundred and seventy-seven) is:
:$29 \times 113$
:The $12$th Poulet number after $341$, $561$, $645$, $1105$, $1387$, $1729$, $1905$, $2047$, $2465$, $2701$, $2821$:
::$2^{3277} \equiv 2 \pmod {3277}$: $3277 = 29 \times 113$
== Also see ==
* 
Category:Specific Numbers
Category:3277
328

== Number ==
$328$ (three hundred and twenty-eight) is:
:$2^3 \times 41$
:The $12$th positive integer after $200$, $202$, $204$, $205$, $206$, $208$, $320$, $322$, $324$, $325$, $326$ that cannot be made into a prime number by changing just $1$ digit
== Also see ==
* 
Category:Specific Numbers
Category:328
328,329

== Number ==
$328 \, 329$ (three hundred and twenty-eight thousand, three hundred and twenty-nine) is:
:$3^2 \times 191^2$
:The $573$rd square number:
::$328 \, 329 = 573 \times 573$
:The $3$rd square number after $8281$, $183 \, 184$ whose digits form $2$ consecutive integers:
::$328 \, 329 = 573^2$
:The $2$nd square number after $183 \, 184$ whose digits form $2$ consecutive increasing integers:
::$328 \, 329 = 573^2$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:328,329
3281

== Number ==
$3281$ (three thousand, two hundred and eighty-one) is:
:$17 \times 193$
:The $15$th Fermat pseudoprime to base $3$ after $91$, $121$, $286$, $671$, $703$, $949$, $1105$, $1541$, $1729$, $1891$, $2465$, $2665$, $2701$, $2821$:
::$3^{3281} \equiv 3 \pmod {3281}$
== Also see ==
* 
Category:Specific Numbers
Category:3281
329

== Number ==
$329$ (three hundred and twenty-nine) is:
:$7 \times 47$
:The $52$nd happy number after $1$, $7$, $10$, $13$, $19$, $23$, $\ldots$, $226$, $230$, $236$, $239$, $262$, $263$, $280$, $291$, $293$, $301$, $302$, $310$, $313$, $319$, $320$, $326$:
::$329 \to 3^2 + 2^2 + 9^2 = 9 + 4 + 36 = 94 \to 9^2 + 4^2 = 81 + 16 = 97 \to 9^2 + 7^2 = 81 + 49 = 130 \to 1^2 + 3^2 + 0^2 = 1 + 9 + 0 = 10 \to 1^2 + 0^2 = 1$
== Also see ==
* 
Category:Specific Numbers
Category:329
33

== Number ==
$33$ (thirty-three) is:
:$3 \times 11$
:The $1$st of the smallest triple of $3$ consecutive semiprimes:
::$33$, $34$, $35$
:The smallest natural number solution to $\map {\sigma_1} n = \map {\sigma_1} {n + 2}$:
::$\map {\sigma_1} {33} = 48 = \map {\sigma_1} {35}$
:The sum of the first $4$ factorials:
::$33 = 1 + 2 + 6 + 24 = 1! + 2! + 3! + 4!$
:The $5$th palindromic lucky number:
::$1$, $3$, $7$, $9$, $33$, $\ldots$
:The $6$th (and last) integer after $2$, $5$, $8$, $12$, $23$ which cannot be expressed as the sum of distinct triangular numbers
:The $7$th integer after $0$, $1$, $3$, $5$, $7$, $9$ which is palindromic in both decimal and binary:
::$33_{10} = 100 \, 001_2$
:The $9$th integer $m$ such that $m! - 1$ (its factorial minus $1$) is prime:
::$3$, $4$, $6$, $7$, $12$, $14$, $30$, $32$, $33$
:The $10$th lucky number:
::$1$, $3$, $7$, $9$, $13$, $15$, $21$, $25$, $31$, $33$, $\ldots$
:The $11$th semiprime after $4$, $6$, $9$, $10$, $14$, $15$, $21$, $22$, $25$, $26$:
::$33 = 3 \times 11$
:The $11$th, and probably largest, integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $9$, $18$ such that both $2^n$ and $5^n$ have no zeroes in their decimal representation:
::$2^{33} = 8 \, 589 \, 934 \, 592$, $5^{33} = 116 \, 415 \, 321 \, 826 \, 934 \, 814 \, 453 \, 125$
:The $12$th and largest (strictly) positive integer after $1$, $2$, $3$, $4$, $6$, $7$, $9$, $10$, $12$, $15$, $18$ which cannot be expressed as the sum of exactly $5$ non-zero squares.
:The $15$th integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $9$, $10$, $11$, $17$, $18$, $30$ such that $5^n$ contains no zero in its decimal representation:
::$5^{33} = 116 \, 415 \, 321 \, 826 \, 934 \, 814 \, 453 \, 125$
:The $17$th odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
::$1$, $3$, $5$, $7$, $9$, $11$, $13$, $15$, $17$, $19$, $21$, $23$, $25$, $27$, $29$, $31$, $33$, $\ldots$
:The $18$th positive integer which is not the sum of $1$ or more distinct squares:
::$2$, $3$, $6$, $7$, $8$, $11$, $12$, $15$, $18$, $19$, $22$, $23$, $24$, $27$, $28$, $31$, $32$, $33$, $\ldots$
:The $22$nd positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $26$, $29$, $30$, $31$, $32$ which cannot be expressed as the sum of distinct pentagonal numbers
:The $23$rd integer $n$ such that $2^n$ contains no zero in its decimal representation:
::$2^{33} = 8 \, 589 \, 934 \, 592$
== Also see ==
* Smallest Sequence of Three Consecutive Semiprimes
* Integer as Sum of 5 Non-Zero Squares
* 
* 
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== Sources ==
* : $33$
* : $33$
Category:Specific Numbers
Category:33
33,100,200,525

== Number ==
$33 \, 100 \, 200 \, 525$ is:
:$3 \times 5^2 \times 7 \times 29 \times 971 \times 2239$
:The larger of the $2$nd of a set of $3$ amicable pairs whose sum is $64 \, 795 \, 852 \, 800$:
:$\map {\sigma_1} {31 \, 695 \, 652 \, 275} = \map {\sigma_1} {33 \, 100 \, 200 \, 525} = 64 \, 795 \, 852 \, 800 = 31 \, 695 \, 652 \, 275 + 33 \, 100 \, 200 \, 525$
== Also see ==
* 
Category:Specific Numbers
Category:33,100,200,525
33,550,336

== Number ==
$33 \, 550 \, 336$ is:
:$2^{12} \times 8191$
:The $5$th perfect number after $6$, $28$, $496$, $8128$:
::$33 \, 550 \, 336 = \map {\sigma_1} {33 \, 550 \, 336} - 33 \, 550 \, 336 = 4096 \times 8191 = 2^{13 - 1} \paren {2^{13} - 1}$
:The $4096$th hexagonal number:
::$33 \, 550 \, 336 = \ds \sum_{k \mathop = 1}^{4096} \paren {4 k - 3} = 4096 \paren {2 \times 4096 - 1}$
:The $8191$st triangular number:
::$33 \, 550 \, 336 = \ds \sum_{k \mathop = 1}^{8191} k = \dfrac {8191 \times \paren {8191 + 1} } 2$
== Also see ==
* 
* 
* 
== Historical Note ==
== Sources ==
* : $33,550,336$
* : $33,550,336$
Category:Specific Numbers
Category:33,550,336
33,550,336/Historical Note

== Historical Note on $33 \, 550 \, 336$ ==
The earliest record of the perfect nature of $33 \, 550 \, 336$ is in a medieval manuscript dating from $1456$.
When demonstrated in $1536$ that both $511 = 2^9 - 1$ and $2047 = 2^{11} - 1$ are composite, it was confirmed as being the $5$th perfect number.
== Sources ==
* : Preface
* : $33,550,336$
* : $33,550,336$
Category:Historical Notes
33,614

== Number ==
$33 \, 614$ (thirty-three thousand, six hundred and fourteen) is:
:$2 \times 7^5$
:The $1$st term of the $1$st triplet of consecutive positive integers each of which is divisible by a $4$th power greater than $1$:
::$33 \, 614 = 14 \times 7^4$, $33 \, 615 = 415 \times 3^4$, $33 \, 616 = 2101 \times 2^4$
== Also see ==
* 
== Sources ==
* : $33,614$
Category:Specific Numbers
Category:33,614
33,615

== Number ==
$33 \, 615$ (thirty-three thousand, six hundred and fifteen) is:
:$3^4 \times 5 \times 83$
:The $2$nd term of the $1$st triplet of consecutive positive integers each of which is divisible by a $4$th power greater than $1$:
::$33 \, 614 = 14 \times 7^4$, $33 \, 615 = 415 \times 3^4$, $33 \, 616 = 2101 \times 2^4$
== Also see ==
* 
== Sources ==
* : $33,614$
Category:Specific Numbers
Category:33,615
33,616

== Number ==
$33 \, 616$ (thirty-three thousand, six hundred and sixteen) is:
:$2^4 \times 11 \times 191$
:The $3$rd term of the $1$st triplet of consecutive positive integers each of which is divisible by a $4$th power greater than $1$:
::$33 \, 614 = 14 \times 7^4$, $33 \, 615 = 415 \times 3^4$, $33 \, 616 = 2101 \times 2^4$
== Also see ==
* 
== Sources ==
* : $33,614$
Category:Specific Numbers
Category:33,616
33,705

== Number ==
$33 \, 705$ (thirty-three thousand, seven hundred and five) is:
:$2^4 \times 11 \times 191$
:The smallest positive integer of the form $3^2 \times 5 \times 7 \paren {2 n + 3}$ for $n \ge 0$ which is not abundant
== Also see ==
* Abundancy of Integers in form 945 + 630n
== Sources ==
* : $33,705$
Category:Specific Numbers
Category:33,705
33,817,088

== Number ==
$33 \, 817 \, 088$ is:
:$2^9 \times 257^2$
:The smallest even integer whose Euler $\phi$ value is shared by no odd integer:
::$\map \phi {33 \, 817 \, 088} = 16 \, 842 \, 752 = 2^{16} \times 257$
== Also see ==
* Smallest Even Integer whose Euler Phi Value is not the Euler Phi Value of an Odd Integer
== Sources ==
* : $33,817,088$
Category:Specific Numbers
Category:33,817,088
330

== Number ==
$330$ (three hundred and thirty) is:
:$2 \times 3 \times 5 \times 11$
:The $8$th pentatope number after $1$, $5$, $15$, $35$, $70$, $126$, $210$:
::$330 = \ds \sum_{k \mathop = 1}^8 \dfrac {k \paren {k + 1} \paren {k + 2} } 6 = \dfrac {8 \paren {8 + 1} \paren {8 + 2} \paren {8 + 3} } {24}$
:The $15$th pentagonal number after $1$, $5$, $12$, $22$, $35$, $51$, $70$, $92$, $117$, $145$, $176$, $210$, $247$, $287$:
::$330 = \ds \sum_{k \mathop = 1}^{15} \paren {3 k - 2} = \dfrac {15 \paren {3 \times 15 - 1} } 2$
:The $29$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $222$, $247$, $260$, $287$, $301$:
::$330 = \dfrac {15 \paren {3 \times 15 - 1} } 2$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:330
331

== Number ==
$331$ (three hundred and thirty-one) is:
:The $67$th prime number
:The number below which it can be guaranteed that no prime factors of $M - 2^n$ or $M + 2^n$ exist for a certain positive integer $n \in \Z_{\ge 0}$ for any integer $M$.
:The larger of the $3$rd pair of primes whose prime gap is $14$:
::$331 - 317 = 14$
:The $5$th obstinate number after $1$, $127$, $149$, $251$
:The $11$th centered hexagonal number after $1$, $7$, $19$, $37$, $61$, $91$, $127$, $169$, $217$, $271$:
::$331 = 1 + 6 + 12 + 18 + 24 + 30 + 36 + 42 + 48 + 54 + 60 = 11^3 - 10^3$
:The $16$th near-repdigit prime after $101$, $113$, $131$, $151$, $181$, $191$, $199$, $211$, $223$, $227$, $229$, $233$, $277$, $311$, $313$
:The $53$rd happy number after $1$, $7$, $10$, $13$, $19$, $23$, $\ldots$, $226$, $230$, $236$, $239$, $262$, $263$, $280$, $291$, $293$, $301$, $302$, $310$, $313$, $319$, $320$, $326$, $329$:
::$331 \to 3^2 + 3^2 + 1^2 = 9 + 9 + 1 = 19 \to 1^2 + 9^2 = 1 + 81 = 82 \to 8^2 + 2^2 = 64 + 4 = 68 \to 6^2 + 8^2 = 36 + 64 = 100 \to 1^2 + 0^2 + 0^2 = 1$
== Also see ==
* Existence of n such that M - 2^n or M + 2^n has no Prime factors less than 331
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* 
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== Sources ==
* : $331$
* : $331$
Category:Specific Numbers
Category:331
3311

== Number ==
$3311$ (three thousand, three hundred and eleven) is:
:$7 \times 11 \times 43$
:The $21$st square pyramidal number after $1$, $5$, $14$, $30$, $55$, $\ldots$, $1015$, $1240$, $1496$, $1785$, $2109$, $2470$, $2870$:
::$3311 = \ds \sum_{k \mathop = 1}^{21} k^2 = \dfrac {21 \paren {21 + 1} \paren {2 \times 21 + 1} } 6$
== Also see ==
* 
Category:Specific Numbers
Category:3311
332

== Number ==
$332$ (three hundred and thirty-two) is:
:$2^2 \times 83$
:The $1$st of the $10$th pair of consecutive integers which both have $6$ divisors:
::$\map {\sigma_0} {332} = \map {\sigma_0} {333} = 6$
== Also see ==
* 
Category:Specific Numbers
Category:332
3321

== Number ==
$3321$ (three thousand, three hundred and twenty-one) is:
:$3^4 \times 41$
:The total of all the entries in a magic square of order $9$, after $1$, $(10)$, $45$, $136$, $325$, $666$, $1225$, $2080$:
::$3321 = \ds \sum_{k \mathop = 1}^{9^2} k = \dfrac {9^2 \paren {9^2 + 1} } 2$
:The $41$st hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $2145$, $2278$, $2415$, $2556$, $2701$, $2850$, $3003$, $3160$:
::$3321 = \ds \sum_{k \mathop = 1}^{41} \paren {4 k - 3} = 41 \paren {2 \times 41 - 1}$
:The $81$st triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $2701$, $2775$, $2850$, $2926$, $3003$, $3081$, $3160$, $3240$:
::$3321 = \ds \sum_{k \mathop = 1}^{81} k = \dfrac {81 \times \paren {81 + 1} } 2$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:3321
333

== Number ==
$333$ (three hundred and thirty-three) is:
:$3^2 \times 37$
:The $2$nd of the $10$th pair of consecutive integers which both have $6$ divisors:
::$\map {\sigma_0} {332} = \map {\sigma_0} {333} = 6$
== Also see ==
* 
Category:Specific Numbers
Category:333
333,667

== Number ==
$333 \, 667$ (three hundred and thirty-three thousand, six hundred and sixty-seven) is:
:The $28 \, 693$rd prime number
:Its product with certain integers exhibits patterns:
::$333 \, 667 \times 296 = 98 \, 765 \, 432$
::$333 \, 667 \times 1113 = 371 \, 371 \, 371$
::$333 \, 667 \times 2223 = 741 \, 741 \, 741$
:Its square is a sorted number:
::$333 \, 667^2 = 111 \, 333 \, 666 \, 889$
:The $7$th unique period prime after $3$, $11$, $37$, $101$, $9091$, $9901$: its period is $9$:
::$\dfrac 1 {333 \, 667} = 0 \cdotp \dot 00000 \, 299 \dot 7$
== Also see ==
* Properties of Family of 333,667 and Related Numbers
* 333,667 is Unique Period Prime with Period 9
* 
* 
== Sources ==
* : $333,667$
* : $333,667$
Category:Specific Numbers
Category:333,667