A319269 Number of uniform factorizations of n into factors > 1, where a factorization is uniform if all factors appear with the same multiplicity.

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 3, 2, 2, 1, 5, 2, 2, 3, 3, 1, 5, 1, 4, 2, 2, 2, 7, 1, 2, 2, 5, 1, 5, 1, 3, 3, 2, 1, 7, 2, 3, 2, 3, 1, 5, 2, 5, 2, 2, 1, 9, 1, 2, 3, 8, 2, 5, 1, 3, 2, 5, 1, 9, 1, 2, 3, 3, 2, 5, 1, 7, 4, 2, 1, 9, 2, 2, 2

Offset

1,4

Links

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

Formula

a(n) = Sum_{d|A052409(n)} A045778(n^(1/d)).

Example

The a(144) = 17 factorizations:
  (144),
  (2*72), (3*48), (4*36), (6*24), (8*18), (9*16), (12*12),
  (2*3*24), (2*4*18), (2*6*12), (2*8*9), (3*4*12), (3*6*8),
  (2*2*6*6), (2*3*4*6), (3*3*4*4).

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], SameQ@@Length/@Split[#]&]], {n, 100}]

Crossrefs

Cf. A001055, A045778, A047966, A052409, A072774, A296132, A303386, A317710, A317712, A317717, A317718.

Sequence in context: A257091 A351418 A359909 * A320888 A296132 A253557

Adjacent sequences: A319266 A319267 A319268 * A319270 A319271 A319272

Keyword

nonn

Author

Gus Wiseman, Sep 16 2018

Status

approved