A289741 a(n) = Kronecker symbol (-20/n).

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

Offset

0,1

Comments

Period 20: repeat [0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1].

This sequence is one of the three non-principal real Dirichlet characters modulo 20. The other two are Jacobi or Kronecker symbols {(20/n)} (or {(n/20)}) and {((-100)/n)} (A185276).

Note that (Sum_{i=0..19} i*a(i))/(-20) = 2 gives the class number of the imaginary quadratic field Q(sqrt(-5)). The fact Q(sqrt(-5)) has class number 2 implies that Q(sqrt(-5)) is not a unique factorization domain.

Links

Antti Karttunen, Table of n, a(n) for n = 0..10000

Eric Weisstein's World of Mathematics, Kronecker Symbol

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,-1).

Formula

a(n) = 1 for n in A045797; -1 for n in A045798; 0 for n that are not coprime with 20.

Completely multiplicative with a(p) = a(p mod 20) for primes p.

a(n) = A080891(n)*A101455(n).

a(n) = -a(n+10) = -a(-n) for all n in Z.

Multiplicative with a(2) = a(5) = 0, a(p) = (-1)^floor(p/10) otherwise; equivalently: a(n) = (-1)^floor(n/10) if n is coprime to 2*5, 0 otherwise. - M. F. Hasler, Feb 28 2022

Mathematica

Array[KroneckerSymbol[-20, #]&, 100, 0] (* Amiram Eldar, Jan 10 2019 *)

Prog

(PARI) a(n) = kronecker(-20, n)

Crossrefs

Cf. A045797, A045798, A185276.

Moebius transform of A035170.

Cf. A240920 (primes not inert in Q(sqrt(-5))), A139513 (primes decomposing), A003626 (primes remaining inert).

Kronecker symbols {(D/n)} for negative fundamental discriminants D = -3..-47, -67, -163: A102283, A101455, A175629, A188510, A011582, A316569, A011585, this sequence, 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: A353529 A286907 A284851 * A185276 A388072 A391503

Adjacent sequences: A289738 A289739 A289740 * A289742 A289743 A289744

Keyword

sign,mult,easy

Author

Jianing Song, Dec 27 2018

Status

approved