A390614 - OEIS

%I #16 Mar 25 2026 15:55:21

%S 0,1,-1,1,1,0,-1,0,-1,1,0,1,1,1,0,0,1,1,-1,-1,0,0,-1,-1,-1,0,-1,1,0,1,

%T 0,-1,-1,1,-1,0,1,-1,1,1,0,-1,0,-1,1,0,1,1,1,0,0,1,1,-1,-1,0,0,-1,-1,

%U -1,0,-1,1,0,1,0,-1,-1,1,-1,0,1,-1,1,1,0,-1,0,-1,1,0,1,1,1,0,0,1,1,-1,-1,0,0,-1,-1,-1,0,-1,1,0,1,0

%N a(n) = Kronecker symbol (-35/n) = (n/35).

%C The Dirichlet character associated with the imaginary quadratic field Q(sqrt(-35)).

%C Note that (Sum_{i=0..35} i*a(i))/(-35) = 2 gives the class number of the imaginary quadratic field Q(sqrt(-35)).

%H Amiram Eldar, <a href="/A390614/b390614.txt">Table of n, a(n) for n = 0..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ClassNumber.html">Class Number</a>.

%H <a href="/index/Rec#order_24">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,-1,1,0,-1,1,0,-1,1,0,-1,0,1,-1,0,1,-1,0,1,-1,0,1,-1).

%F a(n) = A080891(n) * A175629(n).

%F Completely multiplicative with a(5) = a(7) = 0, a(p) = 1 for primes p == 1, 3, 4, 9, 11, 12, 13, 16, 17, 27, 29, 33 (mod 35), a(p) = -1 for primes p == 2, 6, 8, 18, 19, 22, 23, 24, 26, 31, 32, 34 (mod 35).

%F a(n) = (Product_{1<=k<=17} sin(2*k*Pi/35))/(Product_{1<=k<=17} sin(2*Pi/35)) = (sqrt(35)/2^17) * (Product_{1<=k<=17} sin(2*k*Pi/35)).

%F Sum_{n>=1} a(n)/n = -(Pi/35^(3/2)) * (Sum_{i=0..34} i*a(i)) = 2*Pi/sqrt(35) (Dirichlet class number formula).

%t a[n_] := KroneckerSymbol[-35, n]; Array[a, 101, 0] (* _Amiram Eldar_, Mar 25 2026 *)

%o (PARI) a(n) = kronecker(-35, n)

%Y Moebius transform of A035155.

%Y Cf. A191026 (primes decomposing in Q(sqrt(-35))), A191068 (prime remaining inert).

%Y Kronecker symbols {(D/n)} for negative fundamental discriminants D = -3..-47, -67, -163: A102283, A101455, A175629, A188510, A011582, A316569, A011585, A289741, A011586, A109017, A011588, this sequence, A388073, A388072, A011591, A011592, A011596, A011615.

%Y Kronecker symbols {(D/n)} for positive fundamental discriminants D = 5..41: A080891, A091337, A110161, A011583, A011584, A322829, A322796, A390615, A011587, A391502, A011589, A391503, A011590.

%K sign,easy,mult

%O 0

%A _Jianing Song_, Dec 11 2025