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A035182 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s) + Kronecker(m,p)*p^(-2s))^(-1) for m = -7. 9
1, 2, 0, 3, 0, 0, 1, 4, 1, 0, 2, 0, 0, 2, 0, 5, 0, 2, 0, 0, 0, 4, 2, 0, 1, 0, 0, 3, 2, 0, 0, 6, 0, 0, 0, 3, 2, 0, 0, 0, 0, 0, 2, 6, 0, 4, 0, 0, 1, 2, 0, 0, 2, 0, 0, 4, 0, 4, 0, 0, 0, 0, 1, 7, 0, 0, 2, 0, 0, 0, 2, 4, 0, 4, 0, 0, 2, 0, 2, 0, 1, 0, 0, 0, 0, 4, 0, 8, 0, 0, 0, 6, 0, 0, 0, 0, 0, 2, 2, 3, 0, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 + 5*v^2 + 4*w^2 - 8*v*w - 4*u*v + 2*u*w + v - w. - Michael Somos, Jul 21 2004

Half of the number of integer solutions to x^2 + x*y + 2*y^2 = n. - Michael Somos, Jun 05 2005

Inverse Moebius transform of A175629. - Jianing Song, Sep 07 2018

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..10000

FORMULA

a(n) is multiplicative with a(7^e) = 1, a(p^e) = e + 1 if p == 1, 2, 4 (mod 7), a(p^e) = (1 + (-1)^e) / 2 if p == 3, 5, 6 (mod 7). - Michael Somos, May 28 2005

2 * a(n) = A002652(n) unless n = 0.

EXAMPLE

G.f. = x + 2*x^2 + 3*x^4 + x^7 + 4*x^8 + x^9 + 2*x^11 + 2*x^14 + 5*x^16 + ...

MATHEMATICA

a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ -7, d], { d, Divisors[ n]}]]; (* Michael Somos, Jan 23 2014 *)

a[ n_] := If[ n < 1, 0, Length @ FindInstance[ n == x^2 + x y + 2 y^2, {x, y}, Integers, 10^9] / 2]; (* Michael Somos, Jan 23 2014 *)

a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ -7, #] &]]; (* Michael Somos, Jun 10 2015 *)

PROG

(PARI) {a(n) = my(A, p, e); if( n<0, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; [ !(e%2), 1, e+1] [kronecker( -7, p) + 2]))}; /* Michael Somos, May 28 2005 */

(PARI) {a(n) = if( n<1, 0, qfrep([ 2, 1; 1, 4], n, 1)[n])}; /* Michael Somos, Jun 05 2005 */

(PARI) {a(n) = if( n<1, 0, direuler( p=2, n, 1 / ((1 - X) * (1 - kronecker( -7, p)*X)))[n])}; /* Michael Somos, Jun 05 2005 */

(MAGMA) A := Basis( ModularForms( Gamma1(14), 1), 106); B<q> := (-1 + A[1] + 2*A[2] + 4*A[3] + 6*A[5]) / 2; B; /* Michael Somos, Jun 10 2015 */

CROSSREFS

Cf. A002652.

Moebius transform gives A175629.

Sequence in context: A109362 A085246 A268726 * A280720 A245964 A141700

Adjacent sequences:  A035179 A035180 A035181 * A035183 A035184 A035185

KEYWORD

nonn,mult

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified March 23 12:43 EDT 2019. Contains 321430 sequences. (Running on oeis4.)