A347049 Number of odd-length ordered factorizations of n with integer alternating product.

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 4, 1, 3, 1, 3, 1, 1, 1, 5, 1, 1, 2, 3, 1, 1, 1, 7, 1, 1, 1, 11, 1, 1, 1, 5, 1, 1, 1, 3, 3, 1, 1, 14, 1, 3, 1, 3, 1, 5, 1, 5, 1, 1, 1, 7, 1, 1, 3, 15, 1, 1, 1, 3, 1, 1, 1, 24, 1, 1, 3, 3, 1, 1, 1, 14, 4, 1, 1, 7, 1, 1, 1, 5, 1, 7, 1, 3, 1, 1, 1, 24, 1, 3, 3, 11

Offset: 1

Author

Gus Wiseman, Oct 10 2021

Keywords

• nonn

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)).

Examples

			The a(n) ordered factorizations for n = 2, 8, 12, 16, 24, 32, 36, 48:
  2   8       12      16      24      32          36      48
      2*2*2   2*2*3   2*2*4   2*2*6   2*2*8       2*2*9   2*4*6
              3*2*2   2*4*2   3*2*4   2*4*4       2*3*6   3*2*8
                      4*2*2   4*2*3   4*2*4       2*6*3   3*4*4
                              6*2*2   4*4*2       3*2*6   4*2*6
                                      8*2*2       3*3*4   4*4*3
                                      2*2*2*2*2   3*6*2   6*2*4
                                                  4*3*3   6*4*2
                                                  6*2*3   8*2*3
                                                  6*3*2   12*2*2
                                                  9*2*2   2*2*12
                                                          2*2*2*2*3
                                                          2*2*3*2*2
                                                          3*2*2*2*2
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Positions of 2's appear to be A030078.

Positions of 3's appear to be A054753.

Positions of 1's appear to be A167207.

Allowing non-integer alternating product gives A174726, unordered A339890.

The even-length version is A347048.

The unordered version is A347441, with same reverse version.

The case of partitions is A347444, ranked by A347453.

Allowing any length gives A347463.

A001055 counts factorizations (strict A045778, ordered A074206).

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

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

A339846 counts even-length factorizations, ordered A174725.

A347050 = factorizations with alternating permutation, complement A347706.

A347437 = factorizations with integer alternating product, reverse A347442.

A347438 = factorizations with alternating product 1, on squares A273013.

A347439 = factorizations with integer reciprocal alternating product.

A347446 = partitions with integer alternating product, reverse A347445.

A347457 lists Heinz numbers of partitions with integer alternating product.

A347460 counts possible alternating products of factorizations.

A347708 counts possible alternating products of odd-length factorizations.

Cf. A025047, A035363, A038548, A116406, A347440, A347454, A347456, A347458, A347459, A347464, A347704.

Programs

Mathematica

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

PARI

A347049(n, m=n, ap=1, e=0) = if(1==n,(e%2) && 1==denominator(ap), sumdiv(n, d, if(d>1, A347049(n/d, d, ap * d^((-1)^e), 1-e)))); \\ Antti Karttunen, Jul 28 2024

Formula

a(n) = A347463(n) - A347048(n).

Extensions

Data section extended up to a(100) by Antti Karttunen, Jul 28 2024