A011585 Legendre symbol (n,19).

0, 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, 0, 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, 0, 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(-19)).

Note that (Sum_{i=0..18} i*a(i))/(-19) = 1 gives the class number of the imaginary quadratic field Q(sqrt(-19)), i.e., the corresponding ring of integers Z[(1+sqrt(-19))/2] is a unique factorization domain. (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

Harvey P. Dale, Table of n, a(n) for n = 0..1000

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).

Formula

From Jianing Song, Dec 13 2025: (Start)

a(n) = (Product_{k=1..9} sin(2*k*Pi/19))/(Product_{k=1..9} sin(2*Pi/19)) = (sqrt(19)/2^9) * (Product_{k=1..9} sin(2*k*Pi/19)).

Sum_{n>=1} a(n)/n = -(Pi/19^(3/2)) * (Sum_{i=0..18} i*a(i)) = Pi/sqrt(19) (Dirichlet class number formula). (End)

Completely multiplicative with a(19) = 0, a(p) = 1 if p^9 mod 19 = 1, and a(p) = -1 if p^9 mod 19 = 18. - Amiram Eldar, May 23 2026

Mathematica

JacobiSymbol[Range[0, 80], 19] (* Harvey P. Dale, Feb 13 2017 *)

Crossrefs

Moebius transform of A035171.

Cf. A106863 (primes not inert in Q(sqrt(-19))), A191019 (primes decomposing), A191063 (primes remaining inert).

Kronecker symbols {(D/n)} for negative fundamental discriminants D = -3..-47, -67, -163: A102283, A101455, A175629, A188510, A011582, A316569, this sequence, A289741, A011586, A109017, A011588, 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: A168185 A011583 A011584 * A267084 A354807 A011586

Adjacent sequences: A011582 A011583 A011584 * A011586 A011587 A011588

Keyword

sign,easy,mult

Author

N. J. A. Sloane

Status

approved