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

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.

Cf. A000110, A005117, A008480, A124010, A318360.

Factorial numbers: A000142, A007489, A022559, A027423, A071626, A325272, A325617, A336498.

Sequence in context: A076907 A260785 A153897 * A036052 A355207 A308260

Adjacent sequences: A103771 A103772 A103773 * A103775 A103776 A103777

Keyword

nonn

Author

Reinhard Zumkeller, Feb 15 2005

Extensions

a(17)-a(18) from Amiram Eldar, Sep 30 2019

a(19)-a(31) from Giovanni Resta, Sep 30 2019

Status

approved