A327523 Number of factorizations of the n-th number with distinct prime multiplicities A130091(n) into numbers > 1 with distinct prime multiplicities.

1, 1, 1, 2, 1, 1, 3, 2, 1, 3, 1, 5, 1, 3, 1, 3, 1, 5, 2, 3, 3, 1, 1, 7, 1, 5, 1, 1, 3, 3, 1, 9, 2, 3, 3, 1, 5, 5, 1, 1, 3, 11, 1, 3, 1, 11, 1, 3, 3, 1, 9, 5, 1, 5, 1, 3, 14, 1, 3, 3, 1, 1, 5, 1, 11, 1, 9, 1, 3, 3, 2, 3, 3, 1, 15, 1, 5, 5, 1, 1, 20, 3, 3, 1, 1

Offset

1,4

Comments

A number's prime multiplicities are also called its (unsorted) prime signature.

Links

Table of n, a(n) for n=1..85.

Gus Wiseman, Sequences counting and encoding certain classes of multisets

Example

The a(57) = 14 factorizations of 96 together with the corresponding multiset partitions of {1,1,1,1,1,2}:
  (2*2*2*2*2*3)  {{1}{1}{1}{1}{1}{2}}
  (2*2*2*3*4)    {{1}{1}{1}{2}{11}}
  (2*2*2*12)     {{1}{1}{1}{112}}
  (2*2*3*8)      {{1}{1}{2}{111}}
  (2*2*24)       {{1}{1}{1112}}
  (2*3*4*4)      {{1}{2}{11}{11}}
  (2*3*16)       {{1}{2}{1111}}
  (2*4*12)       {{1}{11}{112}}
  (2*48)         {{1}{11112}}
  (3*4*8)        {{2}{11}{111}}
  (3*32)         {{2}{11111}}
  (4*24)         {{11}{1112}}
  (8*12)         {{111}{112}}
  (96)           {{111112}}

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], UnsameQ@@Last/@FactorInteger[#]&];
Table[Length[facsusing[Rest[y], n]], {n, y}]

Crossrefs

See link for additional cross-references.

Cf. A098859, A112798, A130091, A255231.

Sequence in context: A141110 A325758 A280073 * A190770 A292149 A275676

Adjacent sequences: A327520 A327521 A327522 * A327524 A327525 A327526

Keyword

nonn

Author

Gus Wiseman, Sep 16 2019

Status

approved