This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A228885 #22 Mar 21 2025 18:11:33
%S A228885 1,-1,-2,0,4,-4,-6,0,0,-16,-10,0,12,-36,-2048,0,16,0,-18,0,27648,-100,
%T A228885 -22,0,0,-144,0,0,28,-4194304,-30,0,2048000,-256,-127401984,0,36,-324,
%U A228885 -14155776,0,40,-764411904,-42,0,0,-484,-46,0,0,0,-536870912,0,52,0,-419430400000,0,3057647616,-784,-58,0
%N 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.
%C A228885 Conjecture: If n is squarefree, then (-1)^(n*(n-1)/2)*a(n) > 0.
%C A228885 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.
%C A228885 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.
%H A228885 Zhi-Wei Sun, <a href="/A228885/b228885.txt">Table of n, a(n) for n = 1..500</a>
%H A228885 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1308.2900">On some determinants with Legendre symbol entries</a>, preprint, arXiv:1308.2900 [math.NT], 2013-2019.
%F A228885 Conjectures from _Ridouane Oudra_, Mar 13 2025: (Start)
%F A228885 a(n) = (-1)^floor(n/2)*mu(n)^2*Product_{k=1..n} phi(gcd(n,k)).
%F A228885 a(n) = (-1)^floor(n/2)*mu(n)^2*Product_{d|n} phi(d)^phi(n/d).
%F A228885 a(n) = A057077(n)*A008966(n)*A349741(n)*A000010(n). (End)
%e A228885 a(1) = 1 since 1 + 1 = 2 is relatively prime to 1.
%t A228885 a[n_]:=Det[Table[If[GCD[i+j,n]==1,1,0],{i,1,n},{j,1,n}]]
%t A228885 Table[a[n],{n,1,60}]
%o A228885 (PARI) a(n) = matdet(matrix(n, n, i, j, gcd(n, i+j)==1)); \\ _Michel Marcus_, Aug 25 2021
%Y A228885 Cf. A228860, A228884.
%Y A228885 Cf. A057077, A008966, A349741, A000010.
%K A228885 sign
%O A228885 1,3
%A A228885 _Zhi-Wei Sun_, Sep 06 2013