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A035168 a(n) = Sum_{d|n} Kronecker(-22, d). 3
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 (list; graph; refs; listen; history; text; internal format)
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

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

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. - Michael Somos, May 05 2015

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. - Michael Somos, May 05 2015

G.f.: Sum_{k>0} x^k / (1 - x^k) * Kronecker(-22, k). - Michael Somos, May 05 2015

a(n) = A255647(n) unless n = 0. - Michael Somos, May 05 2015

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. A255647.

Sequence in context: A321448 A334158 A116357 * A255647 A119241 A001878

Adjacent sequences:  A035165 A035166 A035167 * A035169 A035170 A035171

KEYWORD

nonn,mult

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified October 17 07:39 EDT 2021. Contains 348048 sequences. (Running on oeis4.)