A323086 Number of factorizations of n into factors > 1 such that no factor is a power of any other (unequal) factor.

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 6, 2, 2, 2, 4, 1, 5, 1, 3, 2, 2, 2, 9, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 9, 2, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 11, 1, 2, 4, 4, 2, 5, 1, 4, 2, 5, 1, 14, 1, 2, 4, 4, 2, 5, 1, 9, 3, 2, 1, 11, 2, 2

Offset

1,4

Links

Table of n, a(n) for n=1..86.

Example

The a(72) = 14 factorizations:
  (2*2*2*3*3),
  (2*2*2*9), (2*2*3*6),
  (2*2*18), (2*3*12), (2*6*6), (3*3*8), (3*4*6),
  (2*36), (3*24), (4*18), (6*12), (8*9),
  (72).

Mathematica

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];
Table[Length[Select[facs[n], stableQ[Union[#], IntegerQ[Log[#1, #2]]&]&]], {n, 100}]

Crossrefs

Cf. A001055, A001597, A002865, A007916, A052410, A101417, A292886, A303707, A305149, A323053, A323087.

Sequence in context: A338430 A223135 A084113 * A345936 A370814 A347050

Adjacent sequences: A323083 A323084 A323085 * A323087 A323088 A323089

Keyword

nonn

Author

Gus Wiseman, Jan 04 2019

Status

approved