A011586 Legendre symbol (n,23).

0, 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, 0, 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

Offset

0,1

Comments

From Jianing Song, Dec 13 2025: (Start)

The Dirichlet character associated with the imaginary quadratic field Q(sqrt(-23)).

Note that (Sum_{i=0..22} i*a(i))/(-23) = 3 gives the class number of the imaginary quadratic field Q(sqrt(-23)). (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).

Formula

From Jianing Song, Dec 13 2025: (Start)

a(n) = (Product_{k=1..11} sin(2*k*Pi/23))/(Product_{k=1..11} sin(2*Pi/23)) = (sqrt(23)/2^11) * (Product_{k=1..11} sin(2*k*Pi/23)).

Sum_{n>=1} a(n)/n = -(Pi/23^(3/2)) * (Sum_{i=0..22} i*a(i)) = 3*Pi/sqrt(23) (Dirichlet class number formula). (End)

Completely multiplicative with a(23) = 0, a(p) = 1 if p^11 mod 23 = 1, and a(p) = -1 if p^11 mod 23 = 22. - Amiram Eldar, May 23 2026

Mathematica

JacobiSymbol[Range[0, 80], 23] (* Harvey P. Dale, Jan 24 2021 *)

Crossrefs

Moebius transform of A035167.

Cf. A296932 (primes not inert in Q(sqrt(-23))), A191021 (primes decomposing), A191065 (primes remaining inert).

Kronecker symbols {(D/n)} for negative fundamental discriminants D = -3..-47, -67, -163: A102283, A101455, A175629, A188510, A011582, A316569, A011585, A289741, this sequence, 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: A011585 A267084 A354807 * A011587 A011588 A011589

Adjacent sequences: A011583 A011584 A011585 * A011587 A011588 A011589

Keyword

sign,easy,mult

Author

N. J. A. Sloane

Status

approved