

A103775


Number of ways to write n! as product of distinct squarefree numbers.


4



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

1,3


COMMENTS

From Gus Wiseman, Aug 23 2020: (Start)
Also the number of setsystems (sets of sets) whose multiset union is the multiset of prime factors of n!. For example, the a(1) = 1 through a(7) = 3 setsystems (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)


LINKS

Table of n, a(n) for n=1..105.
Index entries for sequences related to factorial numbers.


FORMULA

a(n) = 0 for n > 7;
a(n) = A050326(A000142(n)).
a(n) = [C(2*n,n) mod 2] + {C((n+1)^2,n+3) mod 2} + 2*(C(n^2,n+2) mod 2) +{C((n+10)^4,n+12) mod 2} + 3*{C((n+8)^4,n+10) mod 2}.  Paolo P. Lava, Jan 07 2008


EXAMPLE

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.


MATHEMATICA

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 *)


CROSSREFS

A103774 is the nonstrict version.
A337073 is the version for superprimorials, with nonstrict version A337072.
A001055 counts factorizations.
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 setsystems by total sum.
A076716 counts factorizations of factorials.
A116539 counts setsystems covering an initial interval.
A157612 counts strict factorizations of factorials.
Cf. A000110, A005117, A008480, A089259, A116540, A124010, A318360.
Factorial numbers: A000142, A007489, A022559, A027423, A071626, A325272, A325617, A336498.
Sequence in context: A163577 A132178 A039655 * A331594 A093057 A065334
Adjacent sequences: A103772 A103773 A103774 * A103776 A103777 A103778


KEYWORD

nonn


AUTHOR

Reinhard Zumkeller, Feb 15 2005


STATUS

approved



