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A306598 Determinant of the circulant matrix whose first column corresponds to the divisors of n.

%I A306598 #27 Jan 22 2022 00:08:50
%S A306598 1,-3,-8,49,-24,-960,-48,-3375,676,-8640,-120,-2247392,-168,-34560,
%T A306598 -46080,923521,-288,-28789488,-360,-54867456,-184320,-216000,-528,
%U A306598 -89384770560,15376,-423360,-512000,-438939648,-840,-558786571200,-960,-992436543,-1152000
%N A306598 Determinant of the circulant matrix whose first column corresponds to the divisors of n.
%C A306598 From _Robert Israel_, Mar 06 2019: (Start)
%C A306598 a(n) is divisible by A000203(n).
%C A306598 If n is not a square, a(n) is divisible by A000203(n)*A071324(n).
%C A306598 (End)
%H A306598 Robert Israel, <a href="/A306598/b306598.txt">Table of n, a(n) for n = 1..10000</a>
%H A306598 Wikipedia, <a href="https://en.wikipedia.org/wiki/Circulant_matrix">Circulant matrix</a>
%F A306598 Apparently, a(n) > 0 iff n is a square.
%F A306598 a(p) = p^2 - 1 for any prime number p.
%F A306598 a(p^2) = p^6 - 2*p^3 + 1 for any prime number p.
%F A306598 a(2^k) = A086459(k+1) for any k >= 0.
%F A306598 If p < q are primes, a(p*q) = -(p^4-1)*(q^2-1)^2. - _Robert Israel_, Mar 06 2019
%e A306598 For n = 12:
%e A306598 - the divisors of 12 are: 1, 2, 3, 4, 6, 12,
%e A306598 - the corresponding circulant matrix is:
%e A306598     [ 1 12  6  4  3  2]
%e A306598     [ 2  1 12  6  4  3]
%e A306598     [ 3  2  1 12  6  4]
%e A306598     [ 4  3  2  1 12  6]
%e A306598     [ 6  4  3  2  1 12]
%e A306598     [12  6  4  3  2  1]
%e A306598 - its determinant is -2247392,
%e A306598 - hence, a(12) = -2247392.
%p A306598 f:= proc(n) local F,d; uses numtheory, LinearAlgebra;
%p A306598   F:= sort(convert(divisors(n),list));
%p A306598   d:= nops(F);
%p A306598   Determinant(Matrix(d,d,shape=Circulant[F]))
%p A306598 end proc:
%p A306598 map(f, [$1..100]); # _Robert Israel_, Mar 06 2019
%t A306598 a[n_] := Module[{dd = Divisors[n], m, r}, m = Length[dd]; r = E^(2 Pi I/m); Product[Sum[dd[[j+1]] r^(j k), {j, 0, m-1}], {k, 0, m-1}] // FullSimplify];
%t A306598 Array[a, 100] (* _Jean-François Alcover_, Oct 17 2020 *)
%o A306598 (PARI) a(n) = my (d=divisors(n)); my (m=matrix(#d, #d, i,j, d[1+(i-j)%#d])); return (matdet(m))
%Y A306598 Cf. A027750, A086459, A177894.
%K A306598 sign,look
%O A306598 1,2
%A A306598 _Rémy Sigrist_, Feb 27 2019