A301856 Number of subset-products (greater than 1) of factorizations of n into factors greater than 1.

0, 1, 1, 3, 1, 4, 1, 7, 3, 4, 1, 12, 1, 4, 4, 14, 1, 12, 1, 12, 4, 4, 1, 29, 3, 4, 7, 12, 1, 17, 1, 27, 4, 4, 4, 36, 1, 4, 4, 29, 1, 17, 1, 12, 12, 4, 1, 62, 3, 12, 4, 12, 1, 29, 4, 29, 4, 4, 1, 53, 1, 4, 12, 47, 4, 17, 1, 12, 4, 17, 1, 90, 1, 4, 12, 12, 4, 17

Offset: 1

Author

Gus Wiseman, Mar 27 2018

Keywords

• nonn

Comments

For a finite multiset p of positive integers greater than 1 with product n, a pair (t > 1, p) is defined to be a subset-product if there exists a nonempty submultiset of p with product t.

Examples

			The a(12) = 12 subset-products:
12<=(2*2*3), 6<=(2*2*3), 4<=(2*2*3), 3<=(2*2*3), 2<=(2*2*3),
12<=(2*6),   6<=(2*6),   4<=(3*4),   3<=(3*4),   2<=(2*6),
12<=(3*4),
12<=(12).
The a(16) = 14 subset-products:
16<=(16),
16<=(4*4),
16<=(2*8),     8<=(2*8),     4<=(4*4),     2<=(2*8),
16<=(2*2*4),   8<=(2*2*4),   4<=(2*2*4),   2<=(2*2*4),
16<=(2*2*2*2), 8<=(2*2*2*2), 4<=(2*2*2*2), 2<=(2*2*2*2).
		

Crossrefs

Cf. A001055, A000712, A045778, A108917, A162247, A276024, A281116, A284640, A292886, A301829, A301830.

Programs

Mathematica

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Sum[Length[Union[Times@@@Rest[Subsets[f]]]],{f,facs[n]}],{n,100}]