A327522 Number of factorizations of the n-th prime power A000961(n) into prime powers > 1.

0, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 7, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1

Offset

1,7

Links

Table of n, a(n) for n=1..87.

Gus Wiseman, Sequences counting and encoding certain classes of multisets

Formula

a(n) = A000041(A001222(A000961(n))).

Example

The a(32) = 5 factorizations of 81:
  (3*3*3*3)
  (3*3*9)
  (3*27)
  (9*9)
  (81)
The a(32) = 5 multiset partitions of the prime indices of 81 into constant multisets:
  {{2},{2},{2},{2}}
  {{2},{2},{2,2}}
  {{2},{2,2,2}}
  {{2,2},{2,2}}
  {{2,2,2,2}}

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, #]&]}]];
y=Select[Range[nn], PrimePowerQ];
Table[Length[facsusing[Rest[y], n]], {n, y}]

Crossrefs

See link for additional cross-references.

Cf. A000961, A001055, A001222, A001597, A005117, A007916.

Sequence in context: A248641 A282934 A273619 * A034003 A306722 A168508

Adjacent sequences: A327519 A327520 A327521 * A327523 A327524 A327525

Keyword

nonn

Author

Gus Wiseman, Sep 15 2019

Status

approved