A245485 a(n) = 1 if n is a square, -1 if n is seven times a square, 0 otherwise.

1, 0, 0, 1, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1

Offset

1,1

Links

Antti Karttunen, Table of n, a(n) for n = 1..100489

Formula

a(n) is multiplicative with a(p^e) = (-1)^e if p = 7, a(p^e) = (1 + (-1)^e) / 2 otherwise.

G.f.: (theta_3(q) - theta_3(q^7)) / 2 = Sum_{k>0} x^(k^2) - x^(7*k^2).

Dirichlet g.f.: zeta(2*s) * (1 - 7^(-*s)). [corrected by Amiram Eldar, Oct 24 2023]

Sum_{k=1..n} a(k) ~ c*sqrt(n), where c = 1 - 1/sqrt(7) = 0.6220355... . - Amiram Eldar, Oct 24 2023

Example

G.f. = q + q^4 - q^7 + q^9 + q^16 + q^25 - q^28 + q^36 + q^49 - q^63 + ...

Mathematica

a[ n_] := Boole[ IntegerQ[ Sqrt[ n]]] - Boole[ IntegerQ[ Sqrt[ 7 n]]];
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] - EllipticTheta[ 3, 0, q^7]) / 2, {q, 0, n}];
Table[Which[IntegerQ[Sqrt[n]], 1, IntegerQ[Sqrt[n/7]], -1, True, 0], {n, 120}] (* Harvey P. Dale, May 19 2019 *)

Prog

(PARI) {a(n) = issquare(n) - issquare(7*n)};
(PARI) {a(n) = if( n<1, 0, direuler( p=2, n, if( p==7, 1 - X, 1) / (1 - X^2 ))[n])};
(Magma) Basis( ModularForms( Gamma1(28), 1/2), 86) [2] ;

Crossrefs

Cf. A214293, A214295, A244612.

Sequence in context: A373604 A079978 A164704 * A068429 A285208 A260444

Adjacent sequences: A245482 A245483 A245484 * A245486 A245487 A245488

Keyword

sign,mult

Author

Michael Somos, Jul 23 2014

Extensions

Data section extended up to a(112) by Antti Karttunen, Jul 02 2024

Status

approved