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%I #4 Sep 20 2019 08:58:23
%S 1,1,2,1,3,1,2,1,1,5,2,1,4,1,2,2,7,1,1,1,1,4,2,1,7,1,2,1,4,1,5,1,11,2,
%T 2,1,2,1,2,1,7,1,1,1,4,2,1,1,1,12,2,4,1,2,7,2,1,1,10,1,1,2,15,5,1,4,2,
%U 5,1,1,1,1,1,2,4,2,1,1,12,1,2,1,1,2,2
%N Number of factorizations of A302696(n), the n-th number that is 1, 2, or a nonprime number with pairwise coprime prime indices, into factors > 1 satisfying the same conditions.
%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(59) = 10 factorizations of 120 using the allowed factors, together with the corresponding multiset partitions of {1,1,1,2,3}:
%e (2*2*2*15) {{1},{1},{1},{2,3}}
%e (2*2*30) {{1},{1},{1,2,3}}
%e (2*4*15) {{1},{1,1},{2,3}}
%e (2*6*10) {{1},{1,2},{1,3}}
%e (2*60) {{1},{1,1,2,3}}
%e (4*30) {{1,1},{1,2,3}}
%e (6*20) {{1,2},{1,1,3}}
%e (8*15) {{1,1,1},{2,3}}
%e (10*12) {{1,3},{1,1,2}}
%e (120) {{1,1,1,2,3}}
%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],#==1||CoprimeQ@@primeMS[#]&];
%t Table[Length[facsusing[Rest[y],n]],{n,y}]
%Y See link for additional cross-references.
%Y Cf. A001055, A056239, A112798, A302569, A302696, A304711, A305079, A327392.
%K nonn
%O 1,3
%A _Gus Wiseman_, Sep 19 2019
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