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 A306598 Determinant of the circulant matrix whose first column corresponds to the divisors of n. 1

%I

%S 1,-3,-8,49,-24,-960,-48,-3375,676,-8640,-120,-2247392,-168,-34560,

%T -46080,923521,-288,-28789488,-360,-54867456,-184320,-216000,-528,

%U -89384770560,15376,-423360,-512000,-438939648,-840,-558786571200,-960,-992436543,-1152000

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

%C From _Robert Israel_, Mar 06 2019:

%C a(n) is divisible by A000203(n).

%C If n is not a square, a(n) is divisible by A000203(n)*A071324(n).

%C (End)

%H Robert Israel, <a href="/A306598/b306598.txt">Table of n, a(n) for n = 1..10000</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Circulant_matrix">Circulant matrix</a>

%F Apparently, a(n) > 0 iff n is a square.

%F a(p) = p^2 - 1 for any prime number p.

%F a(p^2) = p^6 - 2*p^3 + 1 for any prime number p.

%F a(2^k) = A086459(k+1) for any k >= 0.

%F If p < q are primes, a(p*q) = -(p^4-1)*(q^2-1)^2. - _Robert Israel_, Mar 06 2019

%e For n = 12:

%e - the divisors of 12 are: 1, 2, 3, 4, 6, 12,

%e - the corresponding circulant matrix is:

%e [ 1 12 6 4 3 2]

%e [ 2 1 12 6 4 3]

%e [ 3 2 1 12 6 4]

%e [ 4 3 2 1 12 6]

%e [ 6 4 3 2 1 12]

%e [12 6 4 3 2 1]

%e - its determinant is -2247392,

%e - hence, a(12) = -2247392.

%p f:= proc(n) local F,d; uses numtheory, LinearAlgebra;

%p F:= sort(convert(divisors(n),list));

%p d:= nops(F);

%p Determinant(Matrix(d,d,shape=Circulant[F]))

%p end proc:

%p map(f, [\$1..100]); # _Robert Israel_, Mar 06 2019

%t 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 Array[a, 100] (* _Jean-François Alcover_, Oct 17 2020 *)

%o (PARI) a(n) = my (d=divisors(n)); my (m=matrix(#d, #d, i,j, d[1+(i-j)%#d])); return (matdet(m))

%Y Cf. A027750, A086459, A177894.

%K sign,look

%O 1,2

%A _Rémy Sigrist_, Feb 27 2019

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Last modified June 20 10:53 EDT 2021. Contains 345162 sequences. (Running on oeis4.)