A388072 - OEIS

%I #17 Mar 25 2026 15:55:25

%S 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,

%T -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,

%U 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

%N a(n) = Kronecker symbol (-10/n).

%C Also a(n) = (-40/n).

%C The Dirichlet character associated with the imaginary quadratic field Q(sqrt(-10)) (discriminant -40).

%C Note that (Sum_{i=0..39} i*a(i))/(-40) = 2 gives the class number of the imaginary quadratic field Q(sqrt(-10)).

%H Amiram Eldar, <a href="/A388072/b388072.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_16">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,1,0,0,0,-1,0,0,0,1,0,0,0,-1).

%F a(n) = A080891(n) * A188510(n).

%F Completely multiplicative with a(2) = a(5) = 0, a(p) = 1 for primes p == 1, 7, 9, 11, 13, 19, 23, 37 (mod 40), a(p) = -1 for primes p == 3, 17, 21, 27, 29, 31, 33, 39 (mod 40).

%F Sum_{n>=1} a(n)/n = -(Pi/40^(3/2)) * (Sum_{i=0..39} i*a(i)) = Pi/sqrt(10) (Dirichlet class number formula).

%t a[n_] := KroneckerSymbol[-10, n]; Array[a, 101, 0] (* _Amiram Eldar_, Mar 25 2026 *)

%o (PARI) a(n) = kronecker(-10, n)

%Y Moebius transform of A035180.

%Y Cf. A293859 (primes not inert in Q(sqrt(-10))), A155488 (primes decomposing), A296925 (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, A390614, A388073, this sequence, 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