A320888 Number of set multipartitions (multisets of sets) of factorizations of n into factors > 1 such that all the parts have the same product.

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, 8, 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, 9, 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)} binomial(A045778(n^(1/d)) + d - 1, d).

Example

The a(144) = 20 set multipartitions:
  (2*3*4*6)    (2*8*9)     (2*72)     (144)
  (2*6)*(2*6)  (3*6*8)     (3*48)
  (2*6)*(3*4)  (2*3*24)    (4*36)
  (3*4)*(3*4)  (2*4*18)    (6*24)
               (2*6*12)    (8*18)
               (3*4*12)    (9*16)
               (12)*(2*6)  (12)*(12)
               (12)*(3*4)

Mathematica

strfacs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[strfacs[n/d], Min@@#>d&]], {d, Rest[Divisors[n]]}]];
Table[With[{g=GCD@@FactorInteger[n][[All, 2]]}, Sum[Binomial[Length[strfacs[n^(1/d)]]+d-1, d], {d, Divisors[g]}]], {n, 100}]

Crossrefs

Cf. A001055, A001970, A045778, A050336, A052409, A089259, A294786, A296132, A319269, A320886, A320887, A320889.

Sequence in context: A351418 A359909 A319269 * A296132 A253557 A326191

Adjacent sequences: A320885 A320886 A320887 * A320889 A320890 A320891

Keyword

nonn

Author

Gus Wiseman, Oct 23 2018

Status

approved