A349741 -id:A349741 - cp's OEIS frontend

A228885 Determinant of the n X n matrix with (i,j)-entry equal to 1 or 0 according as i + j is coprime to n or not.

1, -1, -2, 0, 4, -4, -6, 0, 0, -16, -10, 0, 12, -36, -2048, 0, 16, 0, -18, 0, 27648, -100, -22, 0, 0, -144, 0, 0, 28, -4194304, -30, 0, 2048000, -256, -127401984, 0, 36, -324, -14155776, 0, 40, -764411904, -42, 0, 0, -484, -46, 0, 0, 0, -536870912, 0, 52, 0, -419430400000, 0, 3057647616, -784, -58, 0

Offset: 1

Zhi-Wei Sun, Sep 06 2013

sign

Conjecture: If n is squarefree, then (-1)^(n*(n-1)/2)*a(n) > 0.
When p^2 divides n with p prime, (i + n/p) + j is coprime to n if and only if i + j is coprime to n. So a(n) = 0 if n is not squarefree.
It is easy to show that Phi(n) divides a(n) for any n > 0, where Phi(n) is Euler's totient function. Also, a(p) = (-1)^((p-1)/2)*(p-1) for any odd prime p.

			a(1) = 1 since 1 + 1 = 2 is relatively prime to 1.

Zhi-Wei Sun, Table of n, a(n) for n = 1..500
Zhi-Wei Sun, On some determinants with Legendre symbol entries, preprint, arXiv:1308.2900 [math.NT], 2013-2019.

Cf. A228860, A228884.

Cf. A057077, A008966, A349741, A000010.

a[n_]:=Det[Table[If[GCD[i+j,n]==1,1,0],{i,1,n},{j,1,n}]]
Table[a[n],{n,1,60}]

a(n) = matdet(matrix(n, n, i, j, gcd(n, i+j)==1)); \\ Michel Marcus, Aug 25 2021

Conjectures from Ridouane Oudra, Mar 13 2025: (Start)
a(n) = (-1)^floor(n/2)*mu(n)^2*Product_{k=1..n} phi(gcd(n,k)).
a(n) = (-1)^floor(n/2)*mu(n)^2*Product_{d|n} phi(d)^phi(n/d).
a(n) = A057077(n)*A008966(n)*A349741(n)*A000010(n). (End)

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A228885 Determinant of the n X n matrix with (i,j)-entry equal to 1 or 0 according as i + j is coprime to n or not.

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