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A140782 a(n) = sigma(n) * Kronecker(13, n).

%I A140782 #33 Nov 27 2017 00:36:44
%S A140782 1,-3,4,7,-6,-12,-8,-15,13,18,-12,28,0,24,-24,31,18,-39,-20,-42,-32,
%T A140782 36,24,-60,31,0,40,-56,30,72,-32,-63,-48,-54,48,91,-38,60,0,90,-42,96,
%U A140782 44,-84,-78,-72,-48,124,57,-93,72,0,54,-120,72,120,-80,-90,-60,-168,62,96,-104,127,0,144,-68,126,96,-144
%N A140782 a(n) = sigma(n) * Kronecker(13, n).
%C A140782 In the notation of Parry 1979 page 166, the g.f. is (theta_1 - theta_2) / 2 + theta_3 - theta_4 + theta_5 - theta_6 + theta_7 - theta_8 where theta_k is g.f. for A107497, ..., A107504.
%H A140782 W. R. Parry, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002196476">A negative result on the representation of modular forms by theta series</a>, J. Reine Angew. Math., 310 (1979), 151-170.
%F A140782 a(n) is multiplicative with a(p^e) = (p^(e+1) - 1) / (p - 1) * Kronecker(13, p)^e.
%F A140782 G.f. is a period 1 Fourier series which satisfies f(-1 / (169 t)) = -169 (t/i)^2 f(t) where q = exp(2 Pi i t).
%F A140782 a(13*n) = 0. a(n) = A000203(n) * A011583(n). |a(n)| = A000203(n) unless 13 divides n.
%F A140782 a(n) = (A107497(n) - A107498(n)) / 2 + A107499(n) - A107500(n) + A107501(n) - A107502(n) + A107503(n) - A107504(n).
%e A140782 q - 3*q^2 + 4*q^3 + 7*q^4 - 6*q^5 - 12*q^6 - 8*q^7 - 15*q^8 + 13*q^9 + ...
%t A140782 Table[If[n==0, 0, DivisorSigma[1, n] JacobiSymbol[13, n]], {n, 100}] (* _Indranil Ghosh_, Jul 02 2017 *)
%o A140782 (PARI) {a(n) = if( n==0, 0, sigma(n) * kronecker( 13, n))}
%o A140782 (PARI) {a(n) = local(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; (p^(e+1) - 1) / (p - 1) * kronecker( 13, p)^e)))}
%Y A140782 Cf. A000203, A011583, A107497, A107498, A107499, A107500, A107501, A107502, A107503, A107504.
%K A140782 sign,mult
%O A140782 1,2
%A A140782 _Michael Somos_, Jun 04 2008