A320888 - OEIS

%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