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A321860 Number of primes congruent to 2, 6, 7, 8, 10 modulo 11 and <= n minus number of primes congruent to 1, 3, 4, 5, 9 modulo 11 and <= n.

%I A321860 #13 Nov 19 2023 10:23:51
%S A321860 0,1,0,0,-1,-1,0,0,0,0,0,0,1,1,1,1,2,2,3,3,3,3,2,2,2,2,2,2,3,3,2,2,2,
%T A321860 2,2,2,1,1,1,1,2,2,3,3,3,3,2,2,2,2,2,2,1,1,1,1,1,1,0,0,1,1,1,1,1,1,0,
%U A321860 0,0,0,-1,-1,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2
%N A321860 Number of primes congruent to 2, 6, 7, 8, 10 modulo 11 and <= n minus number of primes congruent to 1, 3, 4, 5, 9 modulo 11 and <= n.
%C A321860 a(n) is the number of primes <= n that are quadratic nonresidues modulo 11 minus the number of primes <= n that are quadratic residues modulo 11.
%C A321860 It seems that there are more negative terms here than in some other sequences mentioned in crossrefs; nevertheless, among the first 10000 terms, only 138 ones are negative.
%C A321860 Please see the comment in A321856 describing "Chebyshev's bias" in the general case.
%H A321860 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chebyshev%27s_bias">Chebyshev's bias</a>
%F A321860 a(n) = -Sum_{primes p<=n} Legendre(p,11) = -Sum_{primes p<=n} Kronecker(-11,p) = -Sum_{primes p<=n} A011582(p).
%e A321860 Below 200, there are 20 primes congruent to 1, 3, 4, 5, 9 modulo 11 and 23 primes congruent to 2, 6, 7, 8, 10 modulo 11, so a(200) = 23 - 20 = 3.
%o A321860 (PARI) a(n) = -sum(i=1, n, isprime(i)*kronecker(-11, i))
%Y A321860 Cf. A112632.
%Y A321860 Let d be a fundamental discriminant.
%Y A321860 Sequences of the form "a(n) = -Sum_{primes p<=n} Kronecker(d,p)" with |d| <= 12: this sequence (d=-11), A320857 (d=-8), A321859 (d=-7), A066520 (d=-4), A321856 (d=-3), A321857 (d=5), A071838 (d=8), A321858 (d=12).
%Y A321860 Sequences of the form "a(n) = -Sum_{i=1..n} Kronecker(d,prime(i))" with |d| <= 12: A321865 (d=-11), A320858 (d=-8), A321864 (d=-7), A038698 (d=-4), A112632 (d=-3), A321862 (d=5), A321861 (d=8), A321863 (d=12).
%K A321860 sign
%O A321860 1,17
%A A321860 _Jianing Song_, Nov 20 2018