A293813 Number of partitions of n into nontrivial divisors of n.
1, 0, 0, 0, 1, 0, 2, 0, 3, 1, 2, 0, 11, 0, 2, 2, 9, 0, 14, 0, 15, 2, 2, 0, 79, 1, 2, 4, 19, 0, 93, 0, 35, 2, 2, 2, 279, 0, 2, 2, 157, 0, 153, 0, 27, 24, 2, 0, 1075, 1, 28, 2, 31, 0, 254, 2, 261, 2, 2, 0, 7025, 0, 2, 31, 201, 2, 320, 0, 39, 2, 301, 0, 12071, 0, 2, 35, 43, 2, 427, 0, 3073
Offset: 0
Keywords
Examples
a(6) = 2 because 6 has 4 divisors {1, 2, 3, 6} among which 2 are nontrivial divisors {2, 3} therefore we have [3, 3] and [2, 2, 2].
Links
- Antti Karttunen, Table of n, a(n) for n = 0..10000 (computed from the b-file of A211110 provided by Alois P. Heinz)
- Index entries for sequences related to partitions
Programs
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Maple
with(numtheory): a:= proc(n) local b, l; l:= sort([(divisors(n) minus {1, n})[]]): b:= proc(m, i) option remember; `if`(m=0, 1, `if`(i<1, 0, b(m, i-1)+`if`(l[i]>m, 0, b(m-l[i], i)))) end; forget(b): b(n, nops(l)) end: seq(a(n), n=0..100); # Alois P. Heinz, Oct 16 2017 -
Mathematica
Table[d = Divisors[n]; Coefficient[Series[Product[1/(1 - Boole[d[[k]] != 1 && d[[k]] != n] x^d[[k]]), {k, Length[d]}], {x, 0, n}], x, n], {n, 0, 80}]
Formula
a(n) = [x^n] Product_{d|n, 1 < d < n} 1/(1 - x^d).
a(n) = A211110(n) - 1 for n > 1.