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

1, 1, 2, 10, 141, 6769, 1298995, 1148840085, 5307091649182, 143026276277298216, 24801104674619158730662, 30190572492693121799801655311, 278937095127086600900558327826721594

Offset: 0

Author

Gus Wiseman, Aug 15 2020

Keywords

• nonn
• more

Comments

The n-th superprimorial is A006939(n) = Product_{i = 1..n} prime(i)^(n - i + 1), which has n! divisors.

Also the number of set multipartitions (multisets of sets) of the multiset of prime factors of the superprimorial A006939(n).

Examples

			The a(1) = 1 through a(3) = 10 factorizations:
    2  2*6    2*6*30
       2*2*3  6*6*10
              2*5*6*6
              2*2*3*30
              2*2*6*15
              2*3*6*10
              2*2*3*5*6
              2*2*2*3*15
              2*2*3*3*10
              2*2*2*3*3*5
The a(1) = 1 through a(3) = 10 set multipartitions:
     {1}  {1}{12}    {1}{12}{123}
          {1}{1}{2}  {12}{12}{13}
                     {1}{1}{12}{23}
                     {1}{1}{2}{123}
                     {1}{2}{12}{13}
                     {1}{3}{12}{12}
                     {1}{1}{1}{2}{23}
                     {1}{1}{2}{2}{13}
                     {1}{1}{2}{3}{12}
                     {1}{1}{1}{2}{2}{3}
		

Crossrefs

A000142 counts divisors of superprimorials.

A022915 counts permutations of the same multiset.

A103774 is the version for factorials instead of superprimorials.

A337073 is the strict case (strict factorizations into squarefree numbers).

A001055 counts factorizations.

A006939 lists superprimorials or Chernoff numbers.

A045778 counts strict factorizations.

A050320 counts factorizations into squarefree numbers.

A050326 counts strict factorizations into squarefree numbers.

A076954 can be used instead of A006939 (cf. A307895, A325337).

A089259 counts set multipartitions of integer partitions.

A116540 counts normal set multipartitions.

A317829 counts factorizations of superprimorials.

A337069 counts strict factorizations of superprimorials.

Cf. A000178, A002110, A027423, A124010, A181818, A303279, A318360, A336417, A337070.

Programs

Mathematica

chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
facsqf[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsqf[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Length[facsqf[chern[n]]],{n,0,3}]

PARI

\\ See A318360 for count.
a(n) = {if(n==0, 1, count(vector(n,i,i)))} \\ Andrew Howroyd, Aug 31 2020

Formula

a(n) = A050320(A006939(n)).

a(n) = A318360(A002110(n)). - Andrew Howroyd, Aug 31 2020

Extensions

a(7)-a(12) from Andrew Howroyd, Aug 31 2020