Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the 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