A103774 - OEIS

%I #18 Aug 23 2020 04:00:26

%S 1,1,2,2,6,10,42,42,82,204,1196,1556,10324,34668,104948,104964,873540,

%T 1309396,11855027,25238220,91193575,453628255,5002616219,5902762219,

%U 21142729523,122981607092,189706055368,547296181656,7291700021313,14330422534833,202498591157970

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

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

%C From _Gus Wiseman_, Aug 20 2020: (Start)

%C 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:

%C   {1}  {12}    {1}{1}{12}    {1}{1}{123}      {1}{1}{12}{123}

%C        {1}{2}  {1}{1}{1}{2}  {1}{12}{13}      {1}{12}{12}{13}

%C                              {1}{1}{1}{23}    {1}{1}{1}{12}{23}

%C                              {1}{1}{2}{13}    {1}{1}{1}{2}{123}

%C                              {1}{1}{3}{12}    {1}{1}{2}{12}{13}

%C                              {1}{1}{1}{2}{3}  {1}{1}{3}{12}{12}

%C                                               {1}{1}{1}{1}{2}{23}

%C                                               {1}{1}{1}{2}{2}{13}

%C                                               {1}{1}{1}{2}{3}{12}

%C                                               {1}{1}{1}{1}{2}{2}{3}

%C (End)

%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>.

%e 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.

%t 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 *)

%Y A103775 is the strict case.

%Y A157612 is the case of superprimorials.

%Y A001055 counts factorizations.

%Y A045778 counts strict factorizations.

%Y A048656 counts squarefree divisors of factorials.

%Y A050320 counts factorizations into squarefree numbers.

%Y A050326 counts strict factorizations into squarefree numbers.

%Y A076716 counts factorizations of factorials.

%Y A089259 counts set multipartitions of integer partitions.

%Y A116540 counts normal set multipartitions.

%Y A157612 counts strict factorizations of factorials.

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

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

%K nonn

%O 1,3

%A _Reinhard Zumkeller_, Feb 15 2005

%E a(17)-a(18) from _Amiram Eldar_, Sep 30 2019

%E a(19)-a(31) from _Giovanni Resta_, Sep 30 2019