A103774 - OEIS

A103774

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

1, 1, 2, 2, 6, 10, 42, 42, 82, 204, 1196, 1556, 10324, 34668, 104948, 104964, 873540, 1309396, 11855027, 25238220, 91193575, 453628255, 5002616219, 5902762219, 21142729523, 122981607092, 189706055368, 547296181656, 7291700021313, 14330422534833, 202498591157970

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OFFSET

1,3

COMMENTS

a(n) = A050320(A000142(n)).

From Gus Wiseman, Aug 20 2020: (Start)

Also the number of set multipartitions (multisets of sets) of the multiset of prime factors of n!. For example, The a(2) = 1 through a(6) = 10 set multipartitions are:

{1} {12} {1}{1}{12} {1}{1}{123} {1}{1}{12}{123}

{1}{2} {1}{1}{1}{2} {1}{12}{13} {1}{12}{12}{13}

{1}{1}{1}{23} {1}{1}{1}{12}{23}

{1}{1}{2}{13} {1}{1}{1}{2}{123}

{1}{1}{3}{12} {1}{1}{2}{12}{13}

{1}{1}{1}{2}{3} {1}{1}{3}{12}{12}

{1}{1}{1}{1}{2}{23}

{1}{1}{1}{2}{2}{13}

{1}{1}{1}{2}{3}{12}

{1}{1}{1}{1}{2}{2}{3}

(End)

LINKS

Table of n, a(n) for n=1..31.

Index entries for sequences related to factorial numbers.

EXAMPLE

n=5, 5! = 1*2*3*4*5 = 120 = 2 * 2 * 2 * 3 * 5: a(5)=#{2*2*2*3*5,2*2*2*15,2*2*6*5,2*2*30,2*2*3*10,2*6*10}=6.

MATHEMATICA

sub[w_, e_] := Block[{v=w}, v[[e]]--; v]; ric[w_, k_] := ric[w, k] = If[Max[w] == 0, 1, Block[{e, s, p = Flatten@ Position[Sign@w, 1]}, s = Select[ Prepend[#, First@p] & /@ Subsets[Rest@p], Total[1/2^#] <= k &]; Sum[ric[sub[w, e], Total[1/2^e]], {e, s}]]]; a[n_] := ric[ Sort[ Last /@ FactorInteger[n!]], 1]; Array[a, 22] (* Giovanni Resta, Sep 30 2019 *)

CROSSREFS

A103775 is the strict case.

A157612 is the case of superprimorials.

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.

A076716 counts factorizations of factorials.

A089259 counts set multipartitions of integer partitions.

A116540 counts normal set multipartitions.

A157612 counts strict factorizations of factorials.