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%I #24 Jan 08 2021 10:54:04
%S 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,
%T 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,
%U 7,1,4,1,7,1,38,1,1,4,4,1
%N a(n) = (A002033(n-1) - A008683(n))/2
%C 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
%C From _Gus Wiseman_, Jan 04 2021: (Start)
%C 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:
%C (8) (12) (24) (30) (32) (36)
%C (2*2*2) (2*2*3) (2*2*6) (2*3*5) (2*2*8) (2*2*9)
%C (2*3*2) (2*3*4) (2*5*3) (2*4*4) (2*3*6)
%C (3*2*2) (2*4*3) (3*2*5) (2*8*2) (2*6*3)
%C (2*6*2) (3*5*2) (4*2*4) (2*9*2)
%C (3*2*4) (5*2*3) (4*4*2) (3*2*6)
%C (3*4*2) (5*3*2) (8*2*2) (3*3*4)
%C (4*2*3) (2*2*2*2*2) (3*4*3)
%C (4*3*2) (3*6*2)
%C (6*2*2) (4*3*3)
%C (6*2*3)
%C (6*3*2)
%C (9*2*2)
%C (End)
%H Mats Granvik, <a href="/A174726/b174726.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = (A002033(n-1) - A008683(n))/2. - _Mats Granvik_, May 26 2017
%F For n > 0, a(n) + A174725(n) = A074206(n). - _Gus Wiseman_, Jan 04 2021
%t ordfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@ordfacs[n/d],{d,Rest[Divisors[n]]}]];
%t Table[Length[Select[ordfacs[n],OddQ@*Length]],{n,100}] (* _Gus Wiseman_, Jan 04 2021 *)
%Y The even version is A174725.
%Y The unordered case is A339890, with even version A339846.
%Y A001055 counts factorizations, with strict case A045778.
%Y A074206 counts ordered factorizations, with strict case A254578.
%Y A251683 counts ordered factorizations by product and length.
%Y A340102 counts odd-length factorizations into odd factors.
%Y Other cases of odd length:
%Y - A024429 counts set partitions of odd length.
%Y - A027193 counts partitions of odd length.
%Y - A067659 counts strict partitions of odd length.
%Y - A089677 counts ordered set partitions of odd length.
%Y - A166444 counts compositions of odd length.
%Y - A332304 counts strict compositions of odd length.
%Y Cf. A002033, A024430, A027187, A050320, A052841, A058695, A160786, A316439.
%K nonn
%O 1,8
%A _Mats Granvik_, Mar 28 2010
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