A337073 Number of strict factorizations of the superprimorial A006939(n) into squarefree numbers > 1.

1, 1, 1, 2, 14, 422, 59433, 43181280, 178025660042, 4550598470020490, 782250333882971717562, 974196106965358319940100513, 9412280190038329162111356578977100, 751537739224674099813783040471383322758327

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). It has n! divisors.

Also the number of strict set multipartitions (sets 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*6*30*210
            2*3*6*10  6*10*30*42
                      2*3*6*30*70
                      2*5*6*30*42
                      2*3*10*30*42
                      2*3*6*10*210
                      2*6*10*15*42
                      2*6*10*21*30
                      2*6*14*15*30
                      3*6*10*14*30
                      2*3*5*6*10*42
                      2*3*5*6*14*30
                      2*3*6*7*10*30
                      2*3*6*10*14*15
The a(1) = 1 through a(3) = 14 set multipartitions:
    {1}  {1}{12}  {1}{12}{123}    {1}{12}{123}{1234}
                  {1}{2}{12}{13}  {12}{13}{123}{124}
                                  {1}{12}{13}{23}{124}
                                  {1}{12}{13}{24}{123}
                                  {1}{12}{14}{23}{123}
                                  {1}{2}{12}{123}{134}
                                  {1}{2}{12}{13}{1234}
                                  {1}{2}{13}{123}{124}
                                  {1}{3}{12}{123}{124}
                                  {2}{12}{13}{14}{123}
                                  {1}{2}{12}{13}{14}{23}
                                  {1}{2}{12}{4}{13}{123}
                                  {1}{2}{3}{12}{13}{124}
                                  {1}{2}{3}{12}{14}{123}
		

Crossrefs

A000142 counts divisors of superprimorials.

A022915 counts permutations of the same multiset.

A103775 is the version for factorials instead of superprimorials.

A337072 is the non-strict version.

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.

A050342 counts strict set multipartitions of integer partitions.

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

A283877 counts non-isomorphic strict set multipartitions.

A317829 counts factorizations of superprimorials.

A337069 counts strict factorizations of superprimorials.

Cf. A000178, A002110, A022559, A103774, A124010, A181818, A318361, A336417, A337070.

Programs

Mathematica

chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
ystfac[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[ystfac[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Length[ystfac[chern[n]]],{n,0,4}]

PARI

\\ See A318361 for count.
a(n) = {if(n==0, 1, count(vector(n, i, i)))} \\ Andrew Howroyd, Sep 01 2020

Formula

a(n) = A050326(A006939(n)).

a(n) = A318361(A002110(n)). - Andrew Howroyd, Sep 01 2020

Extensions

a(7)-a(13) from Andrew Howroyd, Sep 01 2020