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A320888
Number of set multipartitions (multisets of sets) of factorizations of n into factors > 1 such that all the parts have the same product.
3
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
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}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 23 2018
STATUS
approved