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Number of factorizations of the superprimorial A006939(n) into squarefree numbers > 1.
4

%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