A337069 Number of strict factorizations of the superprimorial A006939(n).

1, 1, 3, 34, 1591, 360144, 442349835, 3255845551937, 156795416820025934, 53452979022001011490033, 138542156296245533221812350867, 2914321438328993304235584538307144802, 528454951438415221505169213611461783474874149, 873544754831735539240447436467067438924478174290477803

Offset: 0

Author

Gus Wiseman, Aug 15 2020

Keywords

• nonn

Comments

The n-th superprimorial is A006939(n) = Product_{i = 1..n} prime(i)^(n - i + 1).

Also the number of strict multiset partitions of {1,2,2,3,3,3,...,n}, a multiset with i copies of i for i = 1..n.

Examples

			The a(3) = 34 factorizations:
  2*3*4*15  2*3*60   2*180  360
  2*3*5*12  2*4*45   3*120
  2*3*6*10  2*5*36   4*90
  2*4*5*9   2*6*30   5*72
  3*4*5*6   2*9*20   6*60
            2*10*18  8*45
            2*12*15  9*40
            3*4*30   10*36
            3*5*24   12*30
            3*6*20   15*24
            3*8*15   18*20
            3*10*12
            4*5*18
            4*6*15
            4*9*10
            5*6*12
            5*8*9
		

Crossrefs

A022915 counts permutations of the same multiset.

A157612 is the version for factorials instead of superprimorials.

A317829 is the non-strict version.

A337072 is the non-strict version with squarefree factors.

A337073 is the case with squarefree factors.

A000217 counts prime factors (with multiplicity) of superprimorials.

A001055 counts factorizations.

A006939 lists superprimorials or Chernoff numbers.

A045778 counts strict factorizations.

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

A181818 lists products of superprimorials, with complement A336426.

A322583 counts factorizations into factorials.

Cf. A000142, A000178, A002110, A022559, A027423, A303279, A318286, A322583, A337070, A337071.

Programs

Mathematica

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

PARI

\\ See A318286 for count.
a(n) = {if(n==0, 1, count(vector(n, i, i)))} \\ Andrew Howroyd, Sep 01 2020

Formula

a(n) = A045778(A006939(n)).

a(n) = A318286(A002110(n)). - Andrew Howroyd, Sep 01 2020

Extensions

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