A342087 Number of chains of divisors starting with n and having no adjacent parts x <= y^2.

1, 2, 2, 2, 2, 4, 2, 4, 2, 4, 2, 6, 2, 4, 4, 4, 2, 6, 2, 6, 4, 4, 2, 8, 2, 4, 4, 6, 2, 8, 2, 6, 4, 4, 4, 8, 2, 4, 4, 8, 2, 10, 2, 6, 6, 4, 2, 12, 2, 6, 4, 6, 2, 10, 4, 8, 4, 4, 2, 14, 2, 4, 6, 6, 4, 10, 2, 6, 4, 8, 2, 16, 2, 4, 6, 6, 4, 10, 2, 12, 4, 4, 2, 14

Offset: 1

Author

Gus Wiseman, Mar 05 2021

Keywords

• nonn

Comments

An alternative wording: Number of chains of divisors starting with n and having all adjacent parts x > y^2.

Examples

			The chains for n = 1, 2, 6, 12, 24, 42, 48:
   1    2      6        12        24        42          48
        2/1    6/1      12/1      24/1      42/1        48/1
               6/2      12/2      24/2      42/2        48/2
               6/2/1    12/3      24/3      42/3        48/3
                        12/2/1    24/4      42/6        48/4
                        12/3/1    24/2/1    42/2/1      48/6
                                  24/3/1    42/3/1      48/2/1
                                  24/4/1    42/6/1      48/3/1
                                            42/6/2      48/4/1
                                            42/6/2/1    48/6/1
                                                        48/6/2
                                                        48/6/2/1
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

The restriction to powers of 2 is A018819.

Not requiring strict inferiority gives A067824.

The weakly inferior version is twice A337135.

The case ending with 1 is counted by A342083.

The strictly superior version is A342084.

The weakly superior version is A342085.

The additive version is A342098, or A000929 allowing equality.

A000005 counts divisors, with sum A000203.

A001055 counts factorizations.

A003238 counts chains of divisors summing to n-1, with strict case A122651.

A038548 counts inferior (or superior) divisors.

A056924 counts strictly inferior (or strictly superior) divisors.

A067824 counts strict chains of divisors starting with n.

A074206 counts ordered factorizations.

A167865 counts strict chains of divisors > 1 summing to n.

A253249 counts strict chains of divisors.

A334997 counts chains of divisors of n by length.

Cf. A002033, A006530, A018819, A020639, A337105, A341674, A342086, A342094, A342095, A342097.

Programs

Mathematica

cem[n_]:=Prepend[Prepend[#,n]&/@Join@@cem/@Most[Divisors[n]],{n}];
Table[Length[Select[cem[n],And@@Thread[Divide@@@Partition[#,2,1]>Rest[#]]&]],{n,30}]

Formula

For n > 1, a(n) = 2*A342083(n).