A327523 - OEIS

%I #4 Sep 16 2019 12:39:25

%S 1,1,1,2,1,1,3,2,1,3,1,5,1,3,1,3,1,5,2,3,3,1,1,7,1,5,1,1,3,3,1,9,2,3,

%T 3,1,5,5,1,1,3,11,1,3,1,11,1,3,3,1,9,5,1,5,1,3,14,1,3,3,1,1,5,1,11,1,

%U 9,1,3,3,2,3,3,1,15,1,5,5,1,1,20,3,3,1,1

%N Number of factorizations of the n-th number with distinct prime multiplicities A130091(n) into numbers > 1 with distinct prime multiplicities.

%C A number's prime multiplicities are also called its (unsorted) prime signature.

%H Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vSX9dPMGJhxB8rOknCGvOs6PiyhupdWNpqLsnphdgU6MEVqFBnWugAXidDhwHeKqZe_YnUqYeGOXsOk/pub">Sequences counting and encoding certain classes of multisets</a>

%e The a(57) = 14 factorizations of 96 together with the corresponding multiset partitions of {1,1,1,1,1,2}:

%e   (2*2*2*2*2*3)  {{1}{1}{1}{1}{1}{2}}

%e   (2*2*2*3*4)    {{1}{1}{1}{2}{11}}

%e   (2*2*2*12)     {{1}{1}{1}{112}}

%e   (2*2*3*8)      {{1}{1}{2}{111}}

%e   (2*2*24)       {{1}{1}{1112}}

%e   (2*3*4*4)      {{1}{2}{11}{11}}

%e   (2*3*16)       {{1}{2}{1111}}

%e   (2*4*12)       {{1}{11}{112}}

%e   (2*48)         {{1}{11112}}

%e   (3*4*8)        {{2}{11}{111}}

%e   (3*32)         {{2}{11111}}

%e   (4*24)         {{11}{1112}}

%e   (8*12)         {{111}{112}}

%e   (96)           {{111112}}

%t nn=100;

%t facsusing[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsusing[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];

%t y=Select[Range[nn],UnsameQ@@Last/@FactorInteger[#]&];

%t Table[Length[facsusing[Rest[y],n]],{n,y}]

%Y See link for additional cross-references.

%Y Cf. A098859, A112798, A130091, A255231.

%K nonn

%O 1,4

%A _Gus Wiseman_, Sep 16 2019