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%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
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