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

0,7

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

Formula

a(n) = [x^n] Product_{d|n, 1 < d < n} 1/(1 - x^d).

a(n) = A211110(n) - 1 for n > 1.

Example

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].

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}]

Crossrefs

Cf. A018818, A027750, A070824, A210442, A211110, A293814.

Sequence in context: A051709 A318326 A329646 * A218585 A279507 A054656

Adjacent sequences: A293810 A293811 A293812 * A293814 A293815 A293816

Keyword

nonn

Author

Ilya Gutkovskiy, Oct 16 2017

Status

approved