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partition number
A number  $P(n)$  is as partition number if it is equal to the number of ways in which a set of  $n$  identical objects can be partitioned.

For example,  $P(4)=5$  because 4 objects can be partitioned in 5 ways (1)  $\{\{x,x,x,x\}\}$, (2)  $\{\{x\}, \{x,x,x\}\}$, (3)  $\{\{x, x\}, \{x,x\}\}$, (4)  $\{\{x\}, \{x\}, \{x,x\}\}$, and (5)  $\{\{x\}, \{x\}, \{x\}, \{x\}\}$.

There are many recursive formulas for  $P(n)$. For example,

\[
P(n) =\frac{1}{n}\sum_{k=0}^{n-1}\sigma(n-k)P(k)\,.
\]

Hardy and Ramanujan (1918) obtained the asymptotic approximation

\[ P(n) \approx \frac{1}{4n\sqrt{3}} e^{\pi\sqrt{2n/3}}\,.\]

The first partition numbers are 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604, 6842, 8349, 10143 more terms

Partition numbers can also be... (you may click on names or numbers and on + to get more values)

a-pointer 11 101 aban 11 15 22 + 627 792 abundant 30 42 56 + 15796476 26543660 admirable 30 42 56 1002 614154 alt.fact. 101 alternating 30 56 101 + 8349 10143 amenable 56 77 101 + 851376628 952050665 apocalyptic 1002 3010 3718 + 21637 26015 arithmetic 11 15 22 + 8118264 9289091 astonishing 15 Bell 15 betrothed 1575 binomial 15 56 231 792 brilliant 15 2679689 c.decagonal 11 101 c.heptagonal 22 cake 15 42 176 Catalan 42 Chen 11 101 congruent 15 22 30 + 7089500 8118264 constructible 15 30 Cullen 385 Cunningham 15 101 14883 Curzon 30 1958 8349 + 614154 2679689 cyclic 11 15 77 + 6185689 9289091 D-number 15 d-powerful 135 1255 2679689 de Polignac 17977 21637 23338469 decagonal 297 deficient 11 15 22 + 6185689 9289091 dig.balanced 11 15 42 + 56634173 169229875 double fact. 15 Duffinian 77 385 1255 + 6185689 9289091 eban 30 42 56 economical 11 15 101 + 147273 10619863 emirpimes 15 equidigital 11 15 101 + 147273 10619863 esthetic 56 101 4565 Eulerian 11 evil 15 30 77 + 851376628 952050665 Friedman 1255 gapful 135 176 231 + 12292341831 18440293320 Giuga 30 good prime 11 101 happy 176 490 1575 + 4087968 7089500 Harshad 30 42 135 + 4351078600 4835271870 heptagonal 3010 hexagonal 15 231 hoax 22 627 1255 + 7089500 8118264 iban 11 22 42 + 12310 147273 idoneal 15 22 30 42 385 inconsummate 26015 37338 53174 + 526823 614154 interprime 15 30 42 + 15796476 20506255 Jacobsthal 11 junction 101 3718 12310 + 831820 3087735 Kaprekar 297 katadrome 30 42 Lehmer 15 Lucas 11 lucky 15 135 231 + 4697205 5392783 Lynch-Bell 15 135 2436 magic 15 magnanimous 11 30 56 101 metadrome 15 56 135 modest 627 2436 Moran 42 nialpdrome 11 22 30 42 77 nude 11 15 22 + 2436 8118264 oban 11 15 30 + 385 627 octagonal 176 odious 11 22 42 + 607163746 679903203 palindromic 11 22 77 101 palprime 11 101 pancake 11 22 56 panconsummate 11 15 77 231 pandigital 11 15 135 239943 pentagonal 22 176 pernicious 11 22 42 + 6185689 9289091 Perrin 22 plaindrome 11 15 22 + 135 1255 practical 30 42 56 + 7089500 8118264 prim.abundant 30 42 56 + 614154 1741630 prime 11 101 17977 + 80630964769 228204732751 primorial 30 pronic 30 42 56 Proth 385 pseudoperfect 30 42 56 + 715220 831820 repdigit 11 22 77 repunit 15 Ruth-Aaron 15 77 self 42 176 490 + 431149389 541946240 self-describing 22 semiprime 15 22 77 + 72533807 82010177 sliding 11 101 Smith 22 627 1255 sphenic 30 42 231 + 23338469 64112359 straight-line 135 strobogrammatic 11 101 strong prime 11 101 17977 10619863 super Niven 30 1002 super-d 231 12310 105558 + 4087968 7089500 tau 56 792 5604 + 15796476 541946240 tetrahedral 56 tetranacci 15 56 triangular 15 231 twin 11 101 uban 11 15 22 + 56 77 Ulam 11 77 627 + 1505499 2012558 undulating 101 unprimeable 5604 12310 75175 + 7089500 8118264 untouchable 792 1002 1958 + 12310 831820 wasteful 22 30 42 + 8118264 9289091 Zuckerman 11 15 135 1575 Zumkeller 30 42 56 + 31185 37338 zygodrome 11 22 77