A347456 Number of factorizations of n with alternating product >= 1.

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 2, 2, 1, 2, 1, 4, 1, 1, 1, 6, 1, 1, 1, 3, 1, 2, 1, 2, 2, 1, 1, 6, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 5, 1, 1, 2, 8, 1, 2, 1, 2, 1, 2, 1, 8, 1, 1, 2, 2, 1, 2, 1, 6, 4, 1, 1, 5, 1, 1, 1

Offset: 1

Author

Gus Wiseman, Oct 09 2021

Keywords

• nonn

Comments

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

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

Also the number of factorizations of n with alternating sum >= 0.

Examples

			The a(n) factorizations for n = 4, 16, 24, 36, 60, 64, 96:
  4     16        24      36        60       64            96
  2*2   4*4       2*2*6   6*6       2*5*6    8*8           2*6*8
        2*2*4     2*3*4   2*2*9     3*4*5    2*4*8         3*4*8
        2*2*2*2           2*3*6     2*2*15   4*4*4         4*4*6
                          3*3*4     2*3*10   2*2*16        2*2*24
                          2*2*3*3            2*2*4*4       2*3*16
                                             2*2*2*2*4     2*4*12
                                             2*2*2*2*2*2   2*2*2*2*6
                                                           2*2*2*3*4
		

Crossrefs

The case of partitions is A000041, reverse A344607.

The reverse version is A001055, strict A347705.

Positions of 3's appear to be A065036.

Positions of 1's are 1 and A167171.

The opposite version (<= instead of >=) is A339846.

The strict version (> instead of >=) is A339890, also the odd-length case.

Allowing any integer alternating product gives A347437.

The case of alternating product 1 is A347438, also the even-length case.

Allowing any integer reciprocal alternating product gives A347439.

The complement (< instead of >=) is A347440.

Allowing any integer reverse-alternating product gives A347442.

A038548 counts factorizations with a wiggly permutation.

A045778 counts strict factorizations.

A074206 counts ordered factorizations.

A103919 counts partitions by sum and alternating sum (reverse: A344612).

A119620 counts partitions with alternating product 1.

A347447 counts strict factorizations with alternating product > 1.

Cf. A001700, A028983, A316523, A347441, A347443, A347446, A347448, A347450, A347454, A347463, A347708.

Programs

Mathematica

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

Formula

a(n) = A347438(n) + A347440(n).