A348616 Number of ordered factorizations of n with adjacent equal factors.

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 6, 1, 0, 1, 2, 0, 0, 0, 9, 0, 0, 0, 9, 0, 0, 0, 6, 0, 0, 0, 2, 2, 0, 0, 19, 1, 2, 0, 2, 0, 6, 0, 6, 0, 0, 0, 8, 0, 0, 2, 18, 0, 0, 0, 2, 0, 0, 0, 31, 0, 0, 2, 2, 0, 0, 0, 19, 4, 0, 0, 8, 0, 0

Offset: 1

Author

Gus Wiseman, Nov 08 2021

Keywords

• nonn

Comments

First differs from A348613 at a(24) = 6, A348613(24) = 8.

An ordered factorization of n is a finite sequence of positive integers > 1 with product n.

Examples

			The a(n) ordered factorizations with at least one pair of adjacent equal factors for n = 12, 24, 36, 60:
   2*2*3    2*2*6      6*6        15*2*2
   3*2*2    6*2*2      2*2*9      2*2*15
            2*2*2*3    3*3*4      2*2*3*5
            2*2*3*2    4*3*3      2*2*5*3
            2*3*2*2    9*2*2      3*2*2*5
            3*2*2*2    2*2*3*3    3*5*2*2
                       2*3*3*2    5*2*2*3
                       3*2*2*3    5*3*2*2
                       3*3*2*2
See also examples in A348611.
		

Crossrefs

Positions of 0's are A005117.

Positions of 4's appear to be A030514.

Positions of 2's appear to be A054753.

Positions of 1's appear to be A168363.

The additive version (compositions) is A261983, complement A003242.

Factorizations with a permutation of this type are counted by A333487.

Factorizations without a permutation of this type are counted by A335434.

The complement is counted by A348611.

As compositions these are ranked by A348612, complement A333489.

Dominated by A348613 (non-alternating ordered factorizations).

A001055 counts factorizations, strict A045778, ordered A074206.

A335452 counts anti-run permutations of prime indices, complement A336107.

A339846 counts even-length factorizations.

A339890 counts odd-length factorizations.

Cf. A001250, A025047, A122181, A138364, A347050, A347463, A347706, A348379, A348380, A348382, A348610.

Programs

Mathematica

ordfacs[n_]:=If[n<=1,{{}},Join@@Table[Prepend[#,d]&/@ordfacs[n/d],{d,Rest[Divisors[n]]}]];
antirunQ[y_]:=Length[y]==Length[Split[y]]
Table[Length[Select[ordfacs[n],!antirunQ[#]&]],{n,100}]

Formula

a(n) = A074206(n) - A348611(n).