A035176 - OEIS

%I #29 Nov 17 2023 07:34:34

%S 1,1,2,1,2,2,1,1,3,2,0,2,2,1,4,1,0,3,2,2,2,0,2,2,3,2,4,1,0,4,0,1,0,0,

%T 2,3,0,2,4,2,0,2,0,0,6,2,0,2,1,3,0,2,0,4,0,1,4,0,2,4,2,0,3,1,4,0,0,0,

%U 4,2,2,3,0,0,6,2,0,4,2,2,5,0,2,2,0,0,0,0,0,6,2,2,0,0,4,2,0,1,0,3,2,0,0,2,4

%N a(n) = Sum_{d|n} Kronecker(-14, d).

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

%H Amiram Eldar, <a href="/A035176/b035176.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 Dirichlet g.f.: zeta(s) * L(chi,s) where chi(n) = Kronecker( -14, n).

%F From _Michael Somos_, Jun 22 2011: (Start)

%F Expansion of q * f(q^2) * f(q^7) / (chi(q) * chi(q^14)) + 2 * psi(q^4) * phi(-q^7) * chi(-q^14) / chi(-q^2) - 2 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.

%F a(n) is multiplicative with a(p^e) = (1 - q^e) / (1 - q) where q = p * Kronecker( -14, p). (End)

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/sqrt(14) = 1.679251... . - _Amiram Eldar_, Nov 17 2023

%e x + x^2 + 2*x^3 + x^4 + 2*x^5 + 2*x^6 + x^7 + x^8 + 3*x^9 + 2*x^10 + ...

%t a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ -14, d], { d, Divisors[ n]}]] (* _Michael Somos_, Jun 22 2011 *)

%o (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, kronecker( -14, d)))} /* _Michael Somos_, Jun 22 2011 */

%o (PARI) {a(n) = if( n<1, 0, direuler( p=2, n, 1 / ((1 - X) * (1 - kronecker( -14, p) * X)))[n])} /* _Michael Somos_, Jun 22 2011 */

%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( 2 * (eta(x^7 + A) * eta(x^8 + A))^2 / (eta(x^2 + A) * eta(x^28 + A)) - 2 + x * eta(x + A) * eta(x^4 + A)^4 * eta(x^14 + A)^4 * eta(x^56 + A) / (eta(x^2 + A)^3* eta(x^7 + A) * eta(x^8 + A) * eta(x^28 + A)^3), n))} /* _Michael Somos_, Jun 22 2011 */

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

%K nonn,easy,mult

%O 1,3

%A _N. J. A. Sloane_, Dec 11 1999