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%I #5 Oct 23 2018 20:59:23
%S 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,
%T 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,
%U 2,5,1,9,1,2,3,3,2,5,1,7,4,2,1,9,2,2,2
%N Number of set multipartitions (multisets of sets) of factorizations of n into factors > 1 such that all the parts have the same product.
%F a(n) = Sum_{d|A052409(n)} binomial(A045778(n^(1/d)) + d - 1, d).
%e The a(144) = 20 set multipartitions:
%e (2*3*4*6) (2*8*9) (2*72) (144)
%e (2*6)*(2*6) (3*6*8) (3*48)
%e (2*6)*(3*4) (2*3*24) (4*36)
%e (3*4)*(3*4) (2*4*18) (6*24)
%e (2*6*12) (8*18)
%e (3*4*12) (9*16)
%e (12)*(2*6) (12)*(12)
%e (12)*(3*4)
%t strfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strfacs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
%t Table[With[{g=GCD@@FactorInteger[n][[All,2]]},Sum[Binomial[Length[strfacs[n^(1/d)]]+d-1,d],{d,Divisors[g]}]],{n,100}]
%Y Cf. A001055, A001970, A045778, A050336, A052409, A089259, A294786, A296132, A319269, A320886, A320887, A320889.
%K nonn
%O 1,4
%A _Gus Wiseman_, Oct 23 2018
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