%I #14 Aug 31 2020 19:49:57
%S 1,1,2,10,141,6769,1298995,1148840085,5307091649182,
%T 143026276277298216,24801104674619158730662,
%U 30190572492693121799801655311,278937095127086600900558327826721594
%N Number of factorizations of the superprimorial A006939(n) into squarefree numbers > 1.
%C The n-th superprimorial is A006939(n) = Product_{i = 1..n} prime(i)^(n - i + 1), which has n! divisors.
%C Also the number of set multipartitions (multisets of sets) of the multiset of prime factors of the superprimorial A006939(n).
%F a(n) = A050320(A006939(n)).
%F a(n) = A318360(A002110(n)). - _Andrew Howroyd_, Aug 31 2020
%e The a(1) = 1 through a(3) = 10 factorizations:
%e 2 2*6 2*6*30
%e 2*2*3 6*6*10
%e 2*5*6*6
%e 2*2*3*30
%e 2*2*6*15
%e 2*3*6*10
%e 2*2*3*5*6
%e 2*2*2*3*15
%e 2*2*3*3*10
%e 2*2*2*3*3*5
%e The a(1) = 1 through a(3) = 10 set multipartitions:
%e {1} {1}{12} {1}{12}{123}
%e {1}{1}{2} {12}{12}{13}
%e {1}{1}{12}{23}
%e {1}{1}{2}{123}
%e {1}{2}{12}{13}
%e {1}{3}{12}{12}
%e {1}{1}{1}{2}{23}
%e {1}{1}{2}{2}{13}
%e {1}{1}{2}{3}{12}
%e {1}{1}{1}{2}{2}{3}
%t chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
%t facsqf[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsqf[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
%t Table[Length[facsqf[chern[n]]],{n,0,3}]
%o (PARI) \\ See A318360 for count.
%o a(n) = {if(n==0, 1, count(vector(n,i,i)))} \\ _Andrew Howroyd_, Aug 31 2020
%Y A000142 counts divisors of superprimorials.
%Y A022915 counts permutations of the same multiset.
%Y A103774 is the version for factorials instead of superprimorials.
%Y A337073 is the strict case (strict factorizations into squarefree numbers).
%Y A001055 counts factorizations.
%Y A006939 lists superprimorials or Chernoff numbers.
%Y A045778 counts strict factorizations.
%Y A050320 counts factorizations into squarefree numbers.
%Y A050326 counts strict factorizations into squarefree numbers.
%Y A076954 can be used instead of A006939 (cf. A307895, A325337).
%Y A089259 counts set multipartitions of integer partitions.
%Y A116540 counts normal set multipartitions.
%Y A317829 counts factorizations of superprimorials.
%Y A337069 counts strict factorizations of superprimorials.
%Y Cf. A000178, A002110, A027423, A124010, A181818, A303279, A318360, A336417, A337070.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Aug 15 2020
%E a(7)-a(12) from _Andrew Howroyd_, Aug 31 2020