A035168 a(n) = Sum_{d|n} Kronecker(-22, d).

1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 2, 0, 0, 1, 0, 1, 2, 0, 0, 1, 2, 0, 1, 2, 0, 0, 2, 0, 2, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 2, 1, 0, 2, 2, 0, 1, 1, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 2, 2, 0, 1, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 2, 0, 0, 0, 0, 1

Offset

1,13

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Links

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

Michael Somos, Introduction to Ramanujan theta functions, 2019.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Formula

From Michael Somos, May 05 2015: (Start)

Expansion of (phi(q) * phi(q^22) + phi(q^2) * phi(q^11)) / 2 - 1 in powers of q where phi() is a Ramanujan theta function.

a(n) is multiplicative with a(p^e) = 1 if p = 2 or 11, a(p^e) = e+1 if Kronecker(-22, p) = +1, a(p^e) = (1 + (-1)^e)/2 if Kronecker(-22, p) = -1.

G.f.: Sum_{k>0} x^k / (1 - x^k) * Kronecker(-22, k).

a(n) = A255647(n) unless n = 0. (End)

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(22) = 0.669789... . - Amiram Eldar, Nov 17 2023

Example

G.f. = q + q^2 + q^4 + q^8 + q^9 + q^11 + 2*q^13 + q^16 + q^18 + 2*q^19 + ...

Mathematica

a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ -22, #] &]]; (* Michael Somos, May 05 2015 *)
a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ -22, d], { d, Divisors[ n]}]]; (* Michael Somos, May 05 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^22] + EllipticTheta[ 3, 0, q^2] EllipticTheta[ 3, 0, q^11]) / 2 - 1, {q, 0, n}]; (* Michael Somos, May 05 2015 *)

Prog

(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, kronecker( -22, d)))}; /* Michael Somos, May 05 2015 */
(PARI) {a(n) = if( n<1, 0, direuler(p=2, n, 1 / ((1 - X) * (1 - kronecker( -22, p) * X)))[n])}; /* Michael Somos, May 05 2015 */
(PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2 || p==11, 1, kronecker( -22, p) == 1, e+1, 1-e%2)))}; /* Michael Somos, May 05 2015 */

Crossrefs

Cf. A000122, A000700, A010054, A121373, A255647.

Sequence in context: A321448 A334158 A116357 * A255647 A119241 A001878

Adjacent sequences: A035165 A035166 A035167 * A035169 A035170 A035171

Keyword

nonn,easy,mult

Author

N. J. A. Sloane

Status

approved