A174726 a(n) = (A002033(n-1) - A008683(n))/2

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 4, 1, 4, 1, 1, 1, 10, 1, 1, 2, 4, 1, 7, 1, 8, 1, 1, 1, 13, 1, 1, 1, 10, 1, 7, 1, 4, 4, 1, 1, 24, 1, 4, 1, 4, 1, 10, 1, 10, 1, 1, 1, 22, 1, 1, 4, 16, 1, 7, 1, 4, 1, 7, 1, 38, 1, 1, 4, 4, 1

Offset

1,8

Comments

a(n) is the number of permutation matrices with a negative contribution to the determinant that is the Möbius function. See A174725 for how the determinant is defined. - Mats Granvik, May 26 2017

From Gus Wiseman, Jan 04 2021: (Start)

Also the number of ordered factorizations of n into an odd number of factors > 1. The unordered case is A339890. For example, the a(n) factorizations for n = 8, 12, 24, 30, 32, 36 are:

(8) (12) (24) (30) (32) (36)

(2*2*2) (2*2*3) (2*2*6) (2*3*5) (2*2*8) (2*2*9)

(2*3*2) (2*3*4) (2*5*3) (2*4*4) (2*3*6)

(3*2*2) (2*4*3) (3*2*5) (2*8*2) (2*6*3)

(2*6*2) (3*5*2) (4*2*4) (2*9*2)

(3*2*4) (5*2*3) (4*4*2) (3*2*6)

(3*4*2) (5*3*2) (8*2*2) (3*3*4)

(4*2*3) (2*2*2*2*2) (3*4*3)

(4*3*2) (3*6*2)

(6*2*2) (4*3*3)

(6*2*3)

(6*3*2)

(9*2*2)

(End)

Links

Mats Granvik, Table of n, a(n) for n = 1..10000

Formula

a(n) = (A002033(n-1) - A008683(n))/2. - Mats Granvik, May 26 2017

For n > 0, a(n) + A174725(n) = A074206(n). - Gus Wiseman, Jan 04 2021

Mathematica

ordfacs[n_]:=If[n<=1, {{}}, Join@@Table[(Prepend[#1, d]&)/@ordfacs[n/d], {d, Rest[Divisors[n]]}]];
Table[Length[Select[ordfacs[n], OddQ@*Length]], {n, 100}] (* Gus Wiseman, Jan 04 2021 *)

Crossrefs

The even version is A174725.

The unordered case is A339890, with even version A339846.

A001055 counts factorizations, with strict case A045778.

A074206 counts ordered factorizations, with strict case A254578.

A251683 counts ordered factorizations by product and length.

A340102 counts odd-length factorizations into odd factors.

Other cases of odd length:

- A024429 counts set partitions of odd length.

- A027193 counts partitions of odd length.

- A067659 counts strict partitions of odd length.

- A089677 counts ordered set partitions of odd length.

- A166444 counts compositions of odd length.

- A332304 counts strict compositions of odd length.

Cf. A002033, A024430, A027187, A050320, A052841, A058695, A160786, A316439.

Sequence in context: A122374 A261960 A010121 * A300239 A354364 A299203

Adjacent sequences: A174723 A174724 A174725 * A174727 A174728 A174729

Keyword

nonn

Author

Mats Granvik, Mar 28 2010

Status

approved