A011591 Legendre symbol (n,43).

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, 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, 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(-43)).

Note that (Sum_{i=0..42} i*a(i))/(-43) = 1 gives the class number of the imaginary quadratic field Q(sqrt(-43)), i.e., the corresponding ring of integers Z[(1+sqrt(-43))/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

Paolo Xausa, 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,-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<=21} sin(2*k*Pi/42))/(Prod_{1<=k<=21} sin(2*Pi/43)) = (sqrt(43)/2^21) * (Prod_{1<=k<=21} sin(2*k*Pi/43)).

Sum_{n>=1} a(n)/n = -(Pi/43^(3/2)) * (Sum_{i=0..42} i*a(i)) = Pi/sqrt(43) (Dirichlet class number formula). (End)

Mathematica

JacobiSymbol[Range[0, 100], 43] (* Paolo Xausa, Nov 08 2025 *)

Crossrefs

Moebius transform of A035147.

Cf. A106891 (primes not inert in Q(sqrt(-43))), A191031 (primes decomposing), A184902 (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, A011588, A390614, A388073, A388072, this sequence, 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: A011588 A011589 A011590 * A011592 A011593 A011594

Adjacent sequences: A011588 A011589 A011590 * A011592 A011593 A011594

Keyword

sign,mult,easy

Author

N. J. A. Sloane

Status

approved