A035176 a(n) = Sum_{d|n} Kronecker(-14, d).

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, 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, 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

Offset

1,3

Comments

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

Links

Amiram Eldar, 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

Dirichlet g.f.: zeta(s) * L(chi,s) where chi(n) = Kronecker( -14, n).

From Michael Somos, Jun 22 2011: (Start)

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.

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

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

Example

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

Mathematica

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

Prog

(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, kronecker( -14, d)))} /* Michael Somos, Jun 22 2011 */
(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 */
(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 */

Crossrefs

Cf. A000122, A000700, A010054, A121373.

Sequence in context: A366987 A304093 A238580 * A339258 A360326 A322976

Adjacent sequences: A035173 A035174 A035175 * A035177 A035178 A035179

Keyword

nonn,easy,mult

Author

N. J. A. Sloane, Dec 11 1999

Status

approved