A347705 Number of factorizations of n with reverse-alternating product > 1.

0, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 7, 1, 2, 3, 4, 1, 5, 1, 7, 2, 2, 2, 7, 1, 2, 2, 7, 1, 5, 1, 4, 4, 2, 1, 12, 1, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 8, 2, 5, 1, 4, 2, 5, 1, 16, 1, 2, 4, 4, 2, 5, 1, 12, 3, 2, 1, 11, 2

Offset: 1

Author

Gus Wiseman, Oct 12 2021

Keywords

• nonn

Comments

A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)). The reverse-alternating product is the alternating product of the reversed sequence.

Examples

			The a(n) factorizations for n = 2, 6, 8, 12, 24, 30, 48, 60:
  2   6     8       12      24        30      48          60
      2*3   2*4     2*6     3*8       5*6     6*8         2*30
            2*2*2   3*4     4*6       2*15    2*24        3*20
                    2*2*3   2*12      3*10    3*16        4*15
                            2*2*6     2*3*5   4*12        5*12
                            2*3*4             2*3*8       6*10
                            2*2*2*3           2*4*6       2*5*6
                                              3*4*4       3*4*5
                                              2*2*12      2*2*15
                                              2*2*2*6     2*3*10
                                              2*2*3*4     2*2*3*5
                                              2*2*2*2*3
		

Crossrefs

Positions of 1's are A000430.

The weak version (>= instead of >) is A001055, non-reverse A347456.

The non-reverse version is A339890, strict A347447.

The version for reverse-alternating product 1 is A347438.

Allowing any integer reciprocal alternating product gives A347439.

The even-length case is A347440, also the opposite reverse version.

Allowing any integer rev-alt product gives A347442, non-reverse A347437.

The version for partitions is A347449, non-reverse A347448.

A001055 counts factorizations (strict A045778, ordered A074206).

A038548 counts possible rev-alt products of factorizations, integer A046951.

A103919 counts partitions by sum and alternating sum, reverse A344612.

A292886 counts knapsack factorizations, by sum A293627.

A347707 counts possible integer reverse-alternating products of partitions.

Cf. A028983, A119620, A339846, A347441, A347443, A347450, A347463, A347466.

Programs

Mathematica

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
revaltprod[q_]:=Product[q[[-i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[facs[n],revaltprod[#]>1&]],{n,100}]

Formula

a(n) = A001055(n) - A347438(n).