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A327523 Number of factorizations of the n-th number with distinct prime multiplicities A130091(n) into numbers > 1 with distinct prime multiplicities. 0
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified January 26 08:18 EST 2020. Contains 331278 sequences. (Running on oeis4.)