A347441 Number of odd-length factorizations of n with integer alternating product.

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 5, 1, 2, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 2, 5, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 5, 2, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 6, 1, 2, 2, 4, 1, 1, 1, 2, 1, 1, 1, 7

Offset: 1

Author

Gus Wiseman, Sep 07 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)).

Examples

			The a(n) factorizations for n = 2, 8, 32, 48, 54, 72, 108:
  2   8       32          48          54      72          108
      2*2*2   2*2*8       2*4*6       2*3*9   2*6*6       2*6*9
              2*4*4       3*4*4       3*3*6   3*3*8       3*6*6
              2*2*2*2*2   2*2*12              2*2*18      2*2*27
                          2*2*2*2*3           2*3*12      2*3*18
                                              2*2*2*3*3   3*3*12
                                                          2*2*3*3*3
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

The restriction to powers of 2 is A027193.

Positions of 1's are A167207 = A005117 \/ A001248.

Allowing any alternating product gives A339890.

Allowing even-length factorizations gives A347437.

The even-length instead of odd-length version is A347438.

The additive version is A347444, ranked by A347453.

A038548 counts possible reverse-alternating products of factorizations.

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

A339846 counts even-length factorizations.

A347439 counts factorizations with integer reciprocal alternating product.

A347440 counts factorizations with alternating product < 1.

A347442 counts factorizations with integer reverse-alternating product.

A347456 counts factorizations with alternating product >= 1.

A347463 counts ordered factorizations with integer alternating product.

Cf. A062312, A119620, A236913, A330972, A347445, A347446, A347447, A347451, A347458, A347460.

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],OddQ[Length[#]]&&IntegerQ[altprod[#]]&]],{n,100}]

PARI

A347441(n, m=n, ap=1, e=0) = if(1==n, (e%2)&&1==denominator(ap), sumdiv(n, d, if((d>1)&&(d<=m), A347441(n/d, d, ap * d^((-1)^e), 1-e)))); \\ Antti Karttunen, Oct 22 2023

Formula

a(2^n) = A027193(n).

Extensions

Data section extended up to a(108) by Antti Karttunen, Oct 22 2023