%I #24 Nov 17 2023 07:34:47

%S 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,

%T 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,

%U 0,0,2,1,0,0,0,2,0,0,0,0,1

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

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

%H G. C. Greubel, <a href="/A035168/b035168.txt">Table of n, a(n) for n = 1..10000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.

%F From _Michael Somos_, May 05 2015: (Start)

%F 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.

%F 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.

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

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

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

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

%t a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ -22, #] &]]; (* _Michael Somos_, May 05 2015 *)

%t a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ -22, d], { d, Divisors[ n]}]]; (* _Michael Somos_, May 05 2015 *)

%t 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 *)

%o (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, kronecker( -22, d)))}; /* _Michael Somos_, May 05 2015 */

%o (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 */

%o (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 */

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

%K nonn,easy,mult

%O 1,13

%A _N. J. A. Sloane_