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URL: https://oeis.org/A174725

⇱ A174725 - OEIS

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A174725
a(n) = (A074206(n) + A008683(n))/2.
25
1, 0, 0, 1, 0, 2, 0, 2, 1, 2, 0, 4, 0, 2, 2, 4, 0, 4, 0, 4, 2, 2, 0, 10, 1, 2, 2, 4, 0, 6, 0, 8, 2, 2, 2, 13, 0, 2, 2, 10, 0, 6, 0, 4, 4, 2, 0, 24, 1, 4, 2, 4, 0, 10, 2, 10, 2, 2, 0, 22, 0, 2, 4, 16, 2, 6, 0, 4, 2, 6, 0, 38, 0, 2, 4, 4, 2
OFFSET
1,6
COMMENTS
From Mats Granvik, May 25 2017: (Start)
A074206(n) = A002033(n-1) = a(n) + A174726(n).
A008683(n) = a(n) - A174726(n).
Let m = size of matrix a matrix T, and let T be defined as follows:
T(n,k) = if m = 1 then 1 else if mod(n, k) = 0 then if and(n = k, n = m) then 0 else 1 else if and(n = 1, k = m) then 1 else 0
a(n) is then the number of permutation matrices with a positive contribution in the determinant of matrix T. The determinant of T is equal to the Möbius function A008683, see Mathematica program below for how to compute the determinant.
A174726 is the number of permutation matrices with a negative contribution in the determinant of matrix T.
(End)
From Gus Wiseman, Jan 04 2021: (Start)
Also the number of ordered factorizations of n into an even number of factors > 1. The non-ordered case is A339846. For example, the a(n) factorizations for n = 12, 24, 30, 32, 36 are:
(2*6) (3*8) (5*6) (4*8) (4*9)
(3*4) (4*6) (6*5) (8*4) (6*6)
(4*3) (6*4) (10*3) (16*2) (9*4)
(6*2) (8*3) (15*2) (2*16) (12*3)
(12*2) (2*15) (2*2*2*4) (18*2)
(2*12) (3*10) (2*2*4*2) (2*18)
(2*2*2*3) (2*4*2*2) (3*12)
(2*2*3*2) (4*2*2*2) (2*2*3*3)
(2*3*2*2) (2*3*2*3)
(3*2*2*2) (2*3*3*2)
(3*2*2*3)
(3*2*3*2)
(3*3*2*2)
(End)
LINKS
FORMULA
a(n) = (Mobius transform of a(n)) + (Mobius transform of A174726). - Mats Granvik, Apr 04 2010
From Mats Granvik, May 25 2017: (Start)
This sequence is the Moebius transform of A074206.
a(n) = (A074206(n) + A008683(n))/2.
(End)
G.f. A(x) satisfies: A(x) = x + Sum_{i>=2} Sum_{j>=2} A(x^(i*j)). - Ilya Gutkovskiy, May 11 2019
a(n) = A008683(n) + Sum_{d|n, d<n} a(n). - Chai Wah Wu, Mar 25 2026
MATHEMATICA
(* From Mats Granvik, May 25 2017: (Start) *)
Clear[t, nn]; nn = 77; t[1, 1] = 1; t[n_, k_] := t[n, k] = If[k == 1, Sum[t[n, k + i], {i, 1, n - 1}], If[Mod[n, k] == 0, t[n/k, 1], 0], 0]; Monitor[Table[Sum[If[Mod[n, k] == 0, MoebiusMu[k]*t[n/k, 1], 0], {k, 1, 77}], {n, 1, nn}], n]
(* The Möbius function as a determinant *) Table[Det[Table[Table[If[m == 1, 1, If[Mod[n, k] == 0, If[And[n == k, n == m], 0, 1], If[And[n == 1, k == m], 1, 0]]], {k, 1, m}], {n, 1, m}]], {m, 1, 42}]
(* (End) *)
ordfacs[n_]:=If[n<=1, {{}}, Join@@Table[(Prepend[#1, d]&)/@ordfacs[n/d], {d, Rest[Divisors[n]]}]];
Table[Length[Select[ordfacs[n], EvenQ@*Length]], {n, 100}] (* Gus Wiseman, Jan 04 2021 *)
PROG
(Python)
from functools import lru_cache
from sympy import divisors, mobius
@lru_cache(maxsize=None)
def A174725(n): return sum(A174725(d) for d in
divisors(n, generator=True, proper=True))+mobius(n) # Chai Wah Wu, Mar 25 2026
CROSSREFS
The odd version is A174726.
The unordered version is A339846.
A001055 counts factorizations, with strict case A045778.
A058696 counts partitions of even numbers, ranked by A300061.
A074206 counts ordered factorizations, with strict case A254578.
A251683 counts ordered factorizations by product and length.
Other cases of even length:
- A024430 counts set partitions of even length.
- A027187 counts partitions of even length.
- A034008 counts compositions of even length.
- A052841 counts ordered set partitions of even length.
- A067661 counts strict partitions of even length.
- A332305 counts strict compositions of even length
Sequence in context: A307409 A378216 A070824 * A348537 A071459 A319164
KEYWORD
nonn
AUTHOR
Mats Granvik, Mar 28 2010
EXTENSIONS
References to A002033(n-1) changed to A074206(n) by Antti Karttunen, Nov 23 2024
STATUS
approved