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

1, -3, 4, 7, -6, -12, -8, -15, 13, 18, -12, 28, 0, 24, -24, 31, 18, -39, -20, -42, -32, 36, 24, -60, 31, 0, 40, -56, 30, 72, -32, -63, -48, -54, 48, 91, -38, 60, 0, 90, -42, 96, 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

Offset

1,2

Comments

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.

Links

Table of n, a(n) for n=1..70.

W. R. Parry, A negative result on the representation of modular forms by theta series, J. Reine Angew. Math., 310 (1979), 151-170.

Formula

a(n) is multiplicative with a(p^e) = (p^(e+1) - 1) / (p - 1) * Kronecker(13, p)^e.

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).

a(13*n) = 0. a(n) = A000203(n) * A011583(n). |a(n)| = A000203(n) unless 13 divides n.

a(n) = (A107497(n) - A107498(n)) / 2 + A107499(n) - A107500(n) + A107501(n) - A107502(n) + A107503(n) - A107504(n).

Example

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 + ...

Mathematica

Table[If[n==0, 0, DivisorSigma[1, n] JacobiSymbol[13, n]], {n, 100}] (* Indranil Ghosh, Jul 02 2017 *)

Prog

(PARI) {a(n) = if( n==0, 0, sigma(n) * kronecker( 13, n))}
(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)))}

Crossrefs

Cf. A000203, A011583, A107497, A107498, A107499, A107500, A107501, A107502, A107503, A107504.

Sequence in context: A069192 A076887 A351395 * A284587 A097011 A366992

Adjacent sequences: A140779 A140780 A140781 * A140783 A140784 A140785

Keyword

sign,mult

Author

Michael Somos, Jun 04 2008

Status

approved