A011588 Legendre symbol (n,31).

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

Offset

0,1

Comments

From Jianing Song, Dec 13 2025: (Start)

The Dirichlet character associated with the imaginary quadratic field Q(sqrt(-31)).

Note that (Sum_{i=0..30} i*a(i))/(-31) = 3 gives the class number of the imaginary quadratic field Q(sqrt(-31)). (End)

References

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 68.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..10000

Eric Weisstein's World of Mathematics, Class Number.

Index entries for linear recurrences with constant coefficients, signature (-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1).

Formula

From Jianing Song, Dec 13 2025: (Start)

a(n) = (Prod_{1<=k<=15} sin(2*k*Pi/31))/(Prod_{1<=k<=15} sin(2*Pi/31)) = (sqrt(31)/2^15) * (Prod_{1<=k<=15} sin(2*k*Pi/31)).

Sum_{n>=1} a(n)/n = -(Pi/31^(3/2)) * (Sum_{i=0..30} i*a(i)) = 3*Pi/sqrt(31) (Dirichlet class number formula). (End)

Mathematica

JacobiSymbol[Range[0, 80], 31] (* Harvey P. Dale, Jul 30 2020 *)

Prog

(PARI) a(n)=kronecker(n, 31) \\ Charles R Greathouse IV, Feb 02 2026

Crossrefs

Moebius transform of A035159.

Cf. A191024 (primes decomposing in Q(sqrt(-31))), A191067 (primes remaining inert).

Kronecker symbols {(D/n)} for negative fundamental discriminants D = -3..-47, -67, -163: A102283, A101455, A175629, A188510, A011582, A316569, A011585, A289741, A011586, A109017, this sequence, A390614, A388073, A388072, A011591, A011592, A011596, A011615.

Kronecker symbols {(D/n)} for positive fundamental discriminants D = 5..41: A080891, A091337, A110161, A011583, A011584, A322829, A322796, A390615, A011587, A391502, A011589, A391503, A011590.

Sequence in context: A354807 A011586 A011587 * A011589 A011590 A011591

Adjacent sequences: A011585 A011586 A011587 * A011589 A011590 A011591

Keyword

sign,mult,easy

Author

N. J. A. Sloane

Status

approved