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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

From Gus Wiseman, Aug 23 2020: (Start)

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)

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 non-strict version.

A337073 is the version for superprimorials, with non-strict 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 set-systems by total sum.

A076716 counts factorizations of factorials.

A116539 counts set-systems 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: A357869 A039655 A357882 * A331594 A093057 A065334

Adjacent sequences: A103772 A103773 A103774 * A103776 A103777 A103778

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, Feb 15 2005

STATUS

approved

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Last modified November 29 13:48 EST 2022. Contains 358430 sequences. (Running on oeis4.)