This site is supported by donations to The OEIS Foundation.
Multiplicative partitions
From OeisWiki
|
Set of bags | |
---|---|---|
1 | { { } } | |
2 | { {2} } | |
3 | { {3} } | |
4 | { {4}, {2, 2} } | |
5 | { {5} } | |
6 | { {6}, {2, 3} } | |
7 | { {7} } | |
8 | { {8}, {2, 4}, {2, 2, 2} } | |
9 | { {9}, {3, 3} } | |
10 | { {10}, {2, 5} } | |
11 | { {11} } | |
12 | { {12}, {2, 6}, {3, 4}, {2, 2, 3} } |
n |
n |
Examples giving the set of multiplicative partitions (each multiplicative partition expressed as a bag (multiset) of integers greater than 1) of
n |
- 1 has one multiplicative partition (since the empty product is defined as the multiplicative identity, i.e. 1): { ∅ } = { { } };
-
, wherep
is prime, has one multiplicative partition: { { p} };p - 12 has four multiplicative partitions: { {2, 2, 3}, {2, 6}, {3, 4}, {12} };
- 60 has eleven multiplicative partitions: { {2, 2, 3, 5}, {2, 2, 15}, {2, 3, 10}, {2, 5, 6}, {3, 4, 5}, {2, 30}, {3, 20}, {4, 15}, {5, 12}, {6, 10}, {60} }.
n, n ≥ 2, |
n |
- {2, 3, 4, 2, 2, 5, 6, 2, 3, 7, 8, 2, 4, 2, 2, 2, 9, 3, 3, 10, 2, 5, 11, 12, 2, 6, 3, 4, 2, 2, 3, 13, 14, 2, 7, 15, 3, 5, 16, 2, 8, 4, 4, 2, 2, 4, 2, 2, 2, 2, 17, 18, 2, 9, 3, 6, 2, 3, 3, 19, 20, 2, 10, 4, 5, ...}
Contents
n |
n |
n |
Previous examples, expressed as the set of partitions of the multiset (each multiset partition being a multiset of multisets) of the prime factors (with multiplicity) of
n |
- 1: { { ∅ } } = { { { } } };
- p : { { { p} } };
- 12: { { {2}, {2}, {3} }, { {2}, {2, 3} }, { {3}, {2, 2} }, { {2, 2, 3} } };
- 60: { { {2}, {2}, {3}, {5} }, { {2}, {2}, {3, 5} }, { {2}, {3}, {2, 5} }, { {2}, {5}, {2, 3} }, { {3}, {2, 2}, {5} }, { {2}, {2, 3, 5} }, { {3}, {2, 2, 5} }, { {2, 2}, {3, 5} }, { {5}, {2, 2, 3} }, { {2, 3}, {2, 5} }, { {2, 2, 3, 5} } }.
n |
Multiplicative partition function
A001055 The multiplicative partition function: number of ways of factoringn |
a (1) = 1 |
- {1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 7, 2, 2, 3, 4, 1, 5, 1, 7, 2, 2, 2, 9, 1, 2, 2, 7, 1, 5, 1, 4, 4, 2, 1, 12, 2, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 11, 2, 5, 1, 4, 2, 5, 1, 16, 1, 2, 4, 4, 2, 5, 1, 12, 5, ...}
Multiplicative partition function of squarefree numbers
Since the prime factors of a squarefree integer constitute a set (instead of a multiset in the case of a nonsquarefree integer), the number of multiplicative partitions of a squarefree integer withi |
i |
i |
Bi |
A000110 Bell or exponential numbers: number of ways to partition a set of
n, n ≥ 0, |
- {1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597, 27644437, 190899322, 1382958545, 10480142147, 82864869804, 682076806159, 5832742205057, 51724158235372, 474869816156751, ...}
See also
External links
- Weisstein, Eric W., Unordered Factorization, from MathWorld—A Wolfram Web Resource.