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URL: https://oeis.org/A388073

⇱ A388073 - OEIS

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A388073
a(n) = Kronecker symbol (-39/n) = (n/39).
31
0, 1, 1, 0, 1, 1, 0, -1, 1, 0, 1, 1, 0, 0, -1, 0, 1, -1, 0, -1, 1, 0, 1, -1, 0, 1, 0, 0, -1, -1, 0, -1, 1, 0, -1, -1, 0, -1, -1, 0, 1, 1, 0, 1, 1, 0, -1, 1, 0, 1, 1, 0, 0, -1, 0, 1, -1, 0, -1, 1, 0, 1, -1, 0, 1, 0, 0, -1, -1, 0, -1, 1, 0, -1, -1, 0, -1, -1, 0, 1, 1, 0, 1, 1, 0, -1, 1, 0, 1, 1, 0, 0, -1, 0, 1, -1, 0, -1, 1, 0, 1
OFFSET
0
COMMENTS
The Dirichlet character associated with the imaginary quadratic field Q(sqrt(-39)).
Note that (Sum_{i=0..38} i*a(i))/(-39) = 4 gives the class number of the imaginary quadratic field Q(sqrt(-39)).
LINKS
Eric Weisstein's World of Mathematics, Class Number.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,-1,1,-1,1,0,-1,1,-1,1,-1,0,1,-1,1,-1,0,0,0,1,-1).
FORMULA
a(n) = A102283(n) * A011583(n).
Completely multiplicative with a(3) = a(13) = 0, a(p) = 1 for primes p == 1, 2, 4, 5, 8, 10, 11, 16, 20, 22, 25, 32 (mod 39), a(p) = -1 for primes p == 7, 14, 17, 19, 23, 28, 29, 31, 34, 35, 37, 38 (mod 39).
a(n) = (Product_{1<=k<=19} sin(2*k*Pi/39))/(Product_{1<=k<=19} sin(2*Pi/39)) = (sqrt(39)/2^19) * (Product_{1<=k<=19} sin(2*k*Pi/39)).
Sum_{n>=1} a(n)/n = -(Pi/39^(3/2)) * (Sum_{i=0..38} i*a(i)) = 4*Pi/sqrt(39) (Dirichlet class number formula).
MATHEMATICA
a[n_] := KroneckerSymbol[-39, n]; Array[a, 101, 0] (* Amiram Eldar, Mar 25 2026 *)
PROG
(PARI) a(n) = kronecker(-39, n)
CROSSREFS
Moebius transform of A035151.
Cf. A191029 (primes decomposing in Q(sqrt(-39))), A191070 (prime 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, this sequence, 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: A391502 A284792 A094217 * A280261 A174784 A092220
KEYWORD
sign,easy,mult
AUTHOR
Jianing Song, Dec 11 2025
STATUS
approved