A337071 Number of strict chains of divisors starting with n!.

1, 1, 2, 6, 40, 264, 3776, 40256, 1168000, 34204032, 1107791872, 23233380352, 1486675898368, 38934372315136, 1999103691427840, 132874800979423232, 20506322412604129280, 776179999255323115520, 107455579038104865996800, 4651534843901106606571520, 731092060557632280262082560

Offset: 0

Author

Gus Wiseman, Aug 16 2020

Keywords

• nonn

Examples

			The a(1) = 1 through a(3) = 6 chains:
  1  2    6
     2/1  6/1
          6/2
          6/3
          6/2/1
          6/3/1
The a(4) = 40 chains:
  24  24/1   24/2/1   24/4/2/1   24/8/4/2/1
      24/2   24/3/1   24/6/2/1   24/12/4/2/1
      24/3   24/4/1   24/6/3/1   24/12/6/2/1
      24/4   24/4/2   24/8/2/1   24/12/6/3/1
      24/6   24/6/1   24/8/4/1
      24/8   24/6/2   24/8/4/2
      24/12  24/6/3   24/12/2/1
             24/8/1   24/12/3/1
             24/8/2   24/12/4/1
             24/8/4   24/12/4/2
             24/12/1  24/12/6/1
             24/12/2  24/12/6/2
             24/12/3  24/12/6/3
             24/12/4
             24/12/6
		

Crossrefs

A325617 is the maximal case.

A337070 is the version for superprimorials.

A337074 counts the case with distinct prime multiplicities.

A337105 is the case ending with one.

A000005 counts divisors.

A000142 lists factorial numbers.

A027423 counts divisors of factorial numbers.

A067824 counts chains of divisors starting with n.

A074206 counts chains of divisors from n to 1.

A076716 counts factorizations of factorial numbers.

A253249 counts chains of divisors.

Cf. A001013, A001055, A002033, A022559, A124010, A167865, A336941, A336942.

Programs

Mathematica

chnsc[n_]:=Prepend[Join@@Table[Prepend[#,n]&/@chnsc[d],{d,Most[Divisors[n]]}],{n}];
Table[Length[chnsc[n!]],{n,0,5}]

Formula

a(n) = 2*A337105(n) for n > 1.

a(n) = A067824(n!).

Extensions

a(19)-a(20) from Alois P. Heinz, Aug 23 2020