A326976 Number of factorizations of n into factors > 1 such that every prime factor of n is the GCD of some subset of the factors.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1

Offset

1,8

Links

Table of n, a(n) for n=1..87.

Example

The a(72) = 5 factorizations:
  (3*4*6)
  (2*3*12)
  (2*2*3*6)
  (2*3*3*4)
  (2*2*2*3*3)

Mathematica

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

Crossrefs

Factorizations whose dual is a weak antichain are A326975.

T_1 factorizations (whose dual is a strict antichain) are A327012.

T_0 factorizations (whose dual is strict) are A316978.

Cf. A001055, A326947, A326965, A326972, A326974, A326977, A326979.

Sequence in context: A321167 A190867 A355936 * A117358 A294333 A321125

Adjacent sequences: A326973 A326974 A326975 * A326977 A326978 A326979

Keyword

nonn

Author

Gus Wiseman, Aug 13 2019

Status

approved