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A103775
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Number of ways to write n! as product of distinct squarefree numbers.
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4
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1, 1, 2, 0, 1, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
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OFFSET
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1,3
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COMMENTS
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Also the number of set-systems (sets of sets) whose multiset union is the multiset of prime factors of n!. For example, the a(1) = 1 through a(7) = 3 set-systems (empty columns indicated by dots) are:
0 {1} {1,2} . {1},{1,2},{1,3} . {1},{1,2},{1,3},{1,2,4}
{1},{2} {1},{1,2},{1,4},{1,2,3}
{1},{2},{1,2},{1,3},{1,4}
(End)
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LINKS
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FORMULA
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a(n) = 0 for n > 7;
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EXAMPLE
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n=7, 7! = 1*2*3*4*5*6*7 = 5040 = 2*2*2*2*3*3*5*7: a(7) = #{2*3*6*10*14, 2*6*10*42, 2*6*14*30} = 3.
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MATHEMATICA
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yst[n_]:=yst[n]=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[yst[n/d], Min@@#>d&]], {d, Select[Rest[Divisors[n]], SquareFreeQ]}]];
Table[Length[yst[n!]], {n, 15}] (* Gus Wiseman, Aug 21 2020 *)
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CROSSREFS
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A337073 is the version for superprimorials, with non-strict version A337072.
A045778 counts strict factorizations.
A048656 counts squarefree divisors of factorials.
A050320 counts factorizations into squarefree numbers.
A050326 counts strict factorizations into squarefree numbers.
A050342 counts set-systems by total sum.
A076716 counts factorizations of factorials.
A116539 counts set-systems covering an initial interval.
A157612 counts strict factorizations of factorials.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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