A280285 Number of partitions of n into odd composite numbers (A071904).

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 2, 0, 0, 2, 0, 0, 2, 1, 1, 3, 0, 0, 3, 1, 0, 4, 1, 1, 5, 1, 0, 5, 2, 2, 6, 2, 1, 8, 3, 1, 8, 3, 2, 11, 3, 2, 12, 5, 4, 13, 5, 3, 16, 8, 4, 18, 7, 6, 22, 9, 7, 24, 12, 9, 28, 12, 9, 33, 18, 11, 36, 18, 14, 45, 22, 16, 48, 26, 22, 54, 29, 23, 66, 38

Offset

0,28

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Eric Weisstein's World of Mathematics, Composite Number

Index entries for related partition-counting sequences

Formula

G.f.: ((1 - x)/(1 - x^2))*Product_{k>=1} (1 - x^(2*k))*(1 - x^prime(k))/(1 - x^k).

Example

a(36) = 3 because we have [27, 9], [21, 15] and [9, 9, 9, 9].

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(add(
      `if`(d>1 and d::odd and not isprime(d), d, 0),
       d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 31 2016

Mathematica

nmax = 100; CoefficientList[Series[(1 - x)/(1 - x^2) Product[(1 - x^(2 k)) (1 - x^Prime[k])/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Crossrefs

Cf. A002095, A002808, A023895, A071904, A204389, A280287.

Sequence in context: A216921 A344982 A359240 * A033719 A171608 A307985

Adjacent sequences: A280282 A280283 A280284 * A280286 A280287 A280288

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 31 2016

Status

approved