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5 for example can be partitioned in 7 ways thus:

11111, 2111, 221, 311, 32, 41, 5.

The permutations of these 7 partitions add up to 16 thus:

11111 | =1 permutation |

2111 | =4 permutations |

221 | =3 permutations |

311 | =3 permutations |

32 | =2 permutations |

41 | =2 permutations |

5 | =1 permutation |

A partition is a way of writing an integer n as a sum of positive integers where the order of the parts is not significant.

Number | Partitions | Order counts |

0 | 1 | 1 |

1 | 1 | 1 |

2 | 2 | 2 |

3 | 3 | 4 |

4 | 5 | 8 |

5 | 7 | 16 |

6 | 11 | 32 |

7 | 15 | 64 |

8 | 22 | 128 |

9 | 30 | 256 |

10 | 42 | 512 |

11 | 56 | 1024 |

12 | 77 | 2048 |

13 | 101 | 4096 |

14 | 135 | 8192 |

15 | 176 | 16384 |

16 | 231 | - |

17 | 297 | - |

18 | 385 | - |

19 | 490 | - |

20 | 627 | - |

21 | 792 | - |

22 | 1002 | - |

Starting with 5, every 7th integer is a multiple of 7.

Starting with 6, every 11th integer is a multiple of 11.

Number of partitions of

The number of odd partitions of

The total number of 1s that occur among all unordered partitions of a positive integer is equal to the sum of the numbers of distinct members of those partitions.

SEQUENCE: 0, 1, 3, 6, 12, 20, 35, 54, 86, 128, 192, 275, 399, 556, 780, 1068, 1463, 1965, 2644, 3498, 4630, 6052, 7899, 10206, 13174, 16851, 21522, 27294, 34545, 43453, 54563, 68135, 84927, 105366, 130462, 160876, 198014, 242812, 297201, 362587, 441546, 536104, 649791, 785437, 947812, 1140945, 1371173, 1644136, 1968379, 2351597, 2805218, 3339869, 3970648, 4712040, 5584141, 6606438, 7805507, 9207637 ...

The total number of parts in all partitions of

SEQUENCE: 0, 0, 1, 3, 7, 13, 24, 39, 64, 98, 150, 219, 322, 455, 645, 892, 1232, 1668, 2259, 3008, 4003, 5260, 6897, 8951, 11599, 14893, 19086, 24284, 30827, 38888, 48959, 61293, 76578, 95223, 118152, 145993, 180037, 221175, 271186, 331402, 404208, 491521...

The number of "+" signs needed to write the partitions of

Number of partitions of n into distinct parts >= 2....number of partitions of n into odd parts.

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 27, 32, 38, 46, 54, 64, 76, 89, 104, 122, 142, 165, 192, 222, 256, 296, 340, 390, 448, 512, 585, 668, 760, 864, 982, 1113, 1260, 1426, 1610, 1816, 2048, 2304, 2590, 2910, 3264, 3658, 4097, 4582, 5120, 5718

http://www.research.att.com/~njas/sequences/?q=distinct+parts&sort=0&fmt=0&language=english&go=Search

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