cp's OEIS Frontend

A337072 Number of factorizations of the superprimorial A006939(n) into squarefree numbers > 1.

%I A337072 #14 Aug 31 2020 19:49:57
%S A337072 1,1,2,10,141,6769,1298995,1148840085,5307091649182,
%T A337072 143026276277298216,24801104674619158730662,
%U A337072 30190572492693121799801655311,278937095127086600900558327826721594
%N A337072 Number of factorizations of the superprimorial A006939(n) into squarefree numbers > 1.
%C A337072 The n-th superprimorial is A006939(n) = Product_{i = 1..n} prime(i)^(n - i + 1), which has n! divisors.
%C A337072 Also the number of set multipartitions (multisets of sets) of the multiset of prime factors of the superprimorial A006939(n).
%F A337072 a(n) = A050320(A006939(n)).
%F A337072 a(n) = A318360(A002110(n)). - _Andrew Howroyd_, Aug 31 2020
%e A337072 The a(1) = 1 through a(3) = 10 factorizations:
%e A337072     2  2*6    2*6*30
%e A337072        2*2*3  6*6*10
%e A337072               2*5*6*6
%e A337072               2*2*3*30
%e A337072               2*2*6*15
%e A337072               2*3*6*10
%e A337072               2*2*3*5*6
%e A337072               2*2*2*3*15
%e A337072               2*2*3*3*10
%e A337072               2*2*2*3*3*5
%e A337072 The a(1) = 1 through a(3) = 10 set multipartitions:
%e A337072      {1}  {1}{12}    {1}{12}{123}
%e A337072           {1}{1}{2}  {12}{12}{13}
%e A337072                      {1}{1}{12}{23}
%e A337072                      {1}{1}{2}{123}
%e A337072                      {1}{2}{12}{13}
%e A337072                      {1}{3}{12}{12}
%e A337072                      {1}{1}{1}{2}{23}
%e A337072                      {1}{1}{2}{2}{13}
%e A337072                      {1}{1}{2}{3}{12}
%e A337072                      {1}{1}{1}{2}{2}{3}
%t A337072 chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
%t A337072 facsqf[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsqf[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
%t A337072 Table[Length[facsqf[chern[n]]],{n,0,3}]
%o A337072 (PARI) \\ See A318360 for count.
%o A337072 a(n) = {if(n==0, 1, count(vector(n,i,i)))} \\ _Andrew Howroyd_, Aug 31 2020
%Y A337072 A000142 counts divisors of superprimorials.
%Y A337072 A022915 counts permutations of the same multiset.
%Y A337072 A103774 is the version for factorials instead of superprimorials.
%Y A337072 A337073 is the strict case (strict factorizations into squarefree numbers).
%Y A337072 A001055 counts factorizations.
%Y A337072 A006939 lists superprimorials or Chernoff numbers.
%Y A337072 A045778 counts strict factorizations.
%Y A337072 A050320 counts factorizations into squarefree numbers.
%Y A337072 A050326 counts strict factorizations into squarefree numbers.
%Y A337072 A076954 can be used instead of A006939 (cf. A307895, A325337).
%Y A337072 A089259 counts set multipartitions of integer partitions.
%Y A337072 A116540 counts normal set multipartitions.
%Y A337072 A317829 counts factorizations of superprimorials.
%Y A337072 A337069 counts strict factorizations of superprimorials.
%Y A337072 Cf. A000178, A002110, A027423, A124010, A181818, A303279, A318360, A336417, A337070.
%K A337072 nonn,more
%O A337072 0,3
%A A337072 _Gus Wiseman_, Aug 15 2020
%E A337072 a(7)-a(12) from _Andrew Howroyd_, Aug 31 2020