A337256 Number of strict chains of divisors of n.

2, 4, 4, 8, 4, 12, 4, 16, 8, 12, 4, 32, 4, 12, 12, 32, 4, 32, 4, 32, 12, 12, 4, 80, 8, 12, 16, 32, 4, 52, 4, 64, 12, 12, 12, 104, 4, 12, 12, 80, 4, 52, 4, 32, 32, 12, 4, 192, 8, 32, 12, 32, 4, 80, 12, 80, 12, 12, 4, 176, 4, 12, 32, 128, 12, 52, 4, 32, 12, 52

Offset: 1

Author

Gus Wiseman, Aug 23 2020

Keywords

• nonn

Examples

			The a(n) chains for n = 1, 2, 4, 6, 8 (empty chains shown as 0):
  0  0    0      0      0
  1  1    1      1      1
     2    2      2      2
     2/1  4      3      4
          2/1    6      8
          4/1    2/1    2/1
          4/2    3/1    4/1
          4/2/1  6/1    4/2
                 6/2    8/1
                 6/3    8/2
                 6/2/1  8/4
                 6/3/1  4/2/1
                        8/2/1
                        8/4/1
                        8/4/2
                        8/4/2/1
		

Crossrefs

A067824 is the case of chains starting with n (or ending with 1).

A074206 is the case of chains from n to 1.

A253249 is the nonempty case.

A000005 counts divisors.

A001055 counts factorizations.

A001222 counts prime factors with multiplicity.

A074206 counts chains of divisors from n to 1.

A122651 counts chains of divisors summing to n.

A167865 counts chains of divisors > 1 summing to n.

A334996 appears to count chains of divisors from n to 1 by length.

A337070 counts chains of divisors starting with A006939(n).

A337071 counts chains of divisors starting with n!.

A337255 counts chains of divisors starting with n by length.

Cf. A001221, A002033, A008480, A124010, A251683, A337105, A337107.

Programs

Mathematica

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[stableSets[Divisors[n],!(Divisible[#1,#2]||Divisible[#2,#1])&]],{n,10}]

Formula

a(n) = A253249(n) + 1.