A327520 Number of factorizations of the n-th stable number A316476(n) into stable numbers > 1.

1, 1, 2, 1, 1, 3, 2, 1, 1, 2, 5, 1, 1, 1, 2, 3, 1, 1, 7, 2, 2, 1, 1, 1, 4, 1, 2, 2, 1, 2, 1, 1, 11, 1, 2, 1, 1, 4, 2, 1, 5, 1, 2, 1, 2, 2, 2, 1, 4, 1, 1, 1, 1, 1, 2, 2, 2, 3, 1, 15, 1, 7, 1, 1, 2, 2, 2, 1, 1, 4, 2, 1, 2, 1, 5, 1, 2, 1, 4, 2, 1, 1, 2, 1, 1, 1

Offset: 1

Author

Gus Wiseman, Sep 15 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. A number is stable if its distinct prime indices are pairwise indivisible. Stable numbers are listed in A316476.

Examples

			The a(26) = 4 factorizations of 45 into stable numbers:
  (3*3*5)
  (3*15)
  (5*9)
  (45)
The a(201) = 11 multiset partitions of the prime indices of 495 into stable multisets:
  {{2},{2},{3},{5}}
  {{2},{2},{3,5}}
  {{2},{3},{2,5}}
  {{2},{5},{2,3}}
  {{2},{2,3,5}}
  {{3},{2,2},{5}}
  {{3},{2,2,5}}
  {{2,2},{3,5}}
  {{5},{2,2,3}}
  {{2,3},{2,5}}
  {{2,2,3,5}}
		

Links

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

Crossrefs

See link for additional cross-references.

Cf. A001055, A303362, A305149.

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,#]&]}]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
y=Select[Range[nn],stableQ[PrimePi/@First/@FactorInteger[#],Divisible]&];
Table[Length[facsusing[Rest[y],n]],{n,y}]