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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.
9
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.
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.
Sequence in context: A141110 A325758 A280073 * A190770 A292149 A275676
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 16 2019
STATUS
approved