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

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

Offset: 1

Author

Gus Wiseman, Sep 20 2019

Keywords

• nonn

Comments

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.

Examples

			The a(47) = 11 factorizations of 60 together with the corresponding multiset partitions of {1,1,2,3}:
  (2*2*3*5)  {{1},{1},{2},{3}}
  (2*2*15)   {{1},{1},{2,3}}
  (2*3*10)   {{1},{2},{1,3}}
  (2*5*6)    {{1},{3},{1,2}}
  (2*30)     {{1},{1,2,3}}
  (3*4*5)    {{2},{1,1},{3}}
  (3*20)     {{2},{1,1,3}}
  (4*15)     {{1,1},{2,3}}
  (5*12)     {{3},{1,1,2}}
  (6*10)     {{1,2},{1,3}}
  (60)       {{1,1,2,3}}
		

Links

• Gus Wiseman, Sequences counting and encoding certain classes of multisets

Crossrefs

See link for additional cross-references.

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

Programs

Mathematica

nn=100;
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,#]&]}]];
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
y=Select[Range[nn],PrimeQ[#]||CoprimeQ@@primeMS[#]&];
Table[Length[facsusing[Rest[y],n]],{n,y}]

Formula

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