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%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
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