A327518 - OEIS

%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