A347463 Number of ordered factorizations of n with integer alternating product.

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 4, 1, 1, 1, 7, 1, 4, 1, 4, 1, 1, 1, 6, 2, 1, 3, 4, 1, 1, 1, 11, 1, 1, 1, 18, 1, 1, 1, 6, 1, 1, 1, 4, 4, 1, 1, 20, 2, 4, 1, 4, 1, 6, 1, 6, 1, 1, 1, 8, 1, 1, 4, 26, 1, 1, 1, 4, 1, 1, 1, 35, 1, 1, 4, 4, 1, 1, 1, 20, 7, 1, 1, 8, 1

Offset

1,4

Comments

An ordered factorization of n is a 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)).

Links

Table of n, a(n) for n=1..85.

Formula

a(n) = A347048(n) + A347049(n).

Example

The ordered factorizations for n = 4, 8, 12, 16, 24, 32, 36:
  4     8       12      16        24      32          36
  2*2   4*2     6*2     4*4       12*2    8*4         6*6
        2*2*2   2*2*3   8*2       2*2*6   16*2        12*3
                3*2*2   2*2*4     3*2*4   2*2*8       18*2
                        2*4*2     4*2*3   2*4*4       2*2*9
                        4*2*2     6*2*2   4*2*4       2*3*6
                        2*2*2*2           4*4*2       2*6*3
                                          8*2*2       3*2*6
                                          2*2*4*2     3*3*4
                                          4*2*2*2     3*6*2
                                          2*2*2*2*2   4*3*3
                                                      6*2*3
                                                      6*3*2
                                                      9*2*2
                                                      2*2*3*3
                                                      2*3*3*2
                                                      3*2*2*3
                                                      3*3*2*2

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[Join@@Permutations/@facs[n], IntegerQ[altprod[#]]&]], {n, 100}]

Crossrefs

Positions of 2's are A001248.

Positions of 1's are A005117.

The restriction to powers of 2 is A116406.

The even-length case is A347048

The odd-length case is A347049.

The unordered version is A347437, reciprocal A347439, reverse A347442.

The case of partitions is A347446, reverse A347445, ranked by A347457.

A001055 counts factorizations (strict A045778, ordered A074206).

A046099 counts factorizations with no alternating permutations.

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

A119620 counts partitions with alternating product 1, ranked by A028982.

A273013 counts ordered factorizations of n^2 with alternating product 1.

A339846 counts even-length factorizations, ordered A174725.

A339890 counts odd-length factorizations, ordered A174726.

A347438 counts factorizations with alternating product 1.

A347460 counts possible alternating products of factorizations.

Cf. A025047, A038548, A138364, A347440, A347441, A347453, A347454, A347456, A347458, A347459, A347464, A347705, A347708.

Sequence in context: A316978 A331023 A284345 * A373378 A337619 A183214

Adjacent sequences: A347460 A347461 A347462 * A347464 A347465 A347466

Keyword

nonn

Author

Gus Wiseman, Oct 07 2021

Status

approved