A337105 Number of strict chains of divisors from n! to 1.

1, 1, 1, 3, 20, 132, 1888, 20128, 584000, 17102016, 553895936, 11616690176, 743337949184, 19467186157568, 999551845713920, 66437400489711616, 10253161206302064640, 388089999627661557760, 53727789519052432998400, 2325767421950553303285760, 365546030278816140131041280

Offset: 0

Author

Gus Wiseman, Aug 17 2020

Keywords

• nonn

Examples

			The a(4) = 20 chains:
  24/1  24/2/1   24/4/2/1   24/8/4/2/1
        24/3/1   24/6/2/1   24/12/4/2/1
        24/4/1   24/6/3/1   24/12/6/2/1
        24/6/1   24/8/2/1   24/12/6/3/1
        24/8/1   24/8/4/1
        24/12/1  24/12/2/1
                 24/12/3/1
                 24/12/4/1
                 24/12/6/1
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200

Crossrefs

A325617 is the maximal case.

A336941 is the version for superprimorials.

A337104 counts the case with distinct prime multiplicities.

A337071 is the case not necessarily ending with 1.

A000005 counts divisors.

A000142 lists factorial numbers.

A001055 counts factorizations.

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.

A336423 counts chains using A130091, with maximal case A336569.

A336942 counts chains using A130091 from A006939(n) to 1.

Cf. A001013, A001055, A022559, A124010, A167865, A337070, A337074.

Programs

Maple

b:= proc(n) option remember; 1 +
      add(b(d), d=numtheory[divisors](n) minus {n})
    end:
a:= n-> ceil(b(n!)/2):
seq(a(n), n=0..14);  # Alois P. Heinz, Aug 23 2020

Mathematica

chnsc[n_]:=Prepend[Join@@Table[Prepend[#,n]&/@chnsc[d],{d,DeleteCases[Divisors[n],1|n]}],{n}];
Table[Length[chnsc[n!]],{n,0,5}]

Formula

a(n) = A337071(n)/2 for n > 1.

a(n) = A074206(n!).

Extensions

a(19)-a(20) from Alois P. Heinz, Aug 22 2020