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

1,8

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)).

Links

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

Formula

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

Example

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

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}]

Prog

(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

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.

Sequence in context: A355150 A333915 A293234 * A360617 A102097 A361566

Adjacent sequences: A347438 A347439 A347440 * A347442 A347443 A347444

Keyword

nonn

Author

Gus Wiseman, Sep 07 2021

Extensions

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

Status

approved