cp's OEIS Frontend

A327525 Number of factorizations of A302569(n), the n-th number that is 1, prime, or whose prime indices are pairwise coprime.

%I A327525 #8 Sep 20 2019 21:40:29
%S A327525 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 A327525 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 A327525 7,1,4,2,2,2,19,1,1,5,1,7,2,1,1,5,12,1,4
%N A327525 Number of factorizations of A302569(n), the n-th number that is 1, prime, or whose prime indices are pairwise coprime.
%C A327525 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 A327525 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 A327525 a(n) = A001055(A302569(n)).
%e A327525 The a(47) = 11 factorizations of 60 together with the corresponding multiset partitions of {1,1,2,3}:
%e A327525   (2*2*3*5)  {{1},{1},{2},{3}}
%e A327525   (2*2*15)   {{1},{1},{2,3}}
%e A327525   (2*3*10)   {{1},{2},{1,3}}
%e A327525   (2*5*6)    {{1},{3},{1,2}}
%e A327525   (2*30)     {{1},{1,2,3}}
%e A327525   (3*4*5)    {{2},{1,1},{3}}
%e A327525   (3*20)     {{2},{1,1,3}}
%e A327525   (4*15)     {{1,1},{2,3}}
%e A327525   (5*12)     {{3},{1,1,2}}
%e A327525   (6*10)     {{1,2},{1,3}}
%e A327525   (60)       {{1,1,2,3}}
%t A327525 nn=100;
%t A327525 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 A327525 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A327525 y=Select[Range[nn],PrimeQ[#]||CoprimeQ@@primeMS[#]&];
%t A327525 Table[Length[facsusing[Rest[y],n]],{n,y}]
%Y A327525 See link for additional cross-references.
%Y A327525 Cf. A056239, A112798, A281116, A318721, A302569, A304711, A305079.
%K A327525 nonn
%O A327525 1,4
%A A327525 _Gus Wiseman_, Sep 20 2019