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Number of factorizations of A302569(n), the n-th number that is 1, prime, or whose prime indices are pairwise coprime.
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%I #8 Sep 20 2019 21:40:29

%S 1,1,1,2,1,2,1,3,2,1,4,1,2,2,5,1,1,4,2,1,7,2,4,1,5,1,7,2,2,2,1,2,7,1,

%T 1,4,2,1,12,2,4,1,2,7,2,1,11,1,2,11,5,1,4,2,5,1,1,2,4,2,1,12,2,1,2,2,

%U 7,1,4,2,2,2,19,1,1,5,1,7,2,1,1,5,12,1,4

%N Number of factorizations of A302569(n), the n-th number that is 1, prime, or whose prime indices are pairwise coprime.

%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

%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>

%F a(n) = A001055(A302569(n)).

%e The a(47) = 11 factorizations of 60 together with the corresponding multiset partitions of {1,1,2,3}:

%e (2*2*3*5) {{1},{1},{2},{3}}

%e (2*2*15) {{1},{1},{2,3}}

%e (2*3*10) {{1},{2},{1,3}}

%e (2*5*6) {{1},{3},{1,2}}

%e (2*30) {{1},{1,2,3}}

%e (3*4*5) {{2},{1,1},{3}}

%e (3*20) {{2},{1,1,3}}

%e (4*15) {{1,1},{2,3}}

%e (5*12) {{3},{1,1,2}}

%e (6*10) {{1,2},{1,3}}

%e (60) {{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 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t y=Select[Range[nn],PrimeQ[#]||CoprimeQ@@primeMS[#]&];

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

%Y See link for additional cross-references.

%Y Cf. A056239, A112798, A281116, A318721, A302569, A304711, A305079.

%K nonn

%O 1,4

%A _Gus Wiseman_, Sep 20 2019