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A306500 a(n) is the smallest prime p such that Sum_{primes q <= p} Kronecker(-A003657(n),q) > 0, or 0 if no such prime exists. 5
608981813029, 26861, 2, 3, 5, 2, 11, 3, 2, 5, 2, 11, 2, 11, 53, 2, 13, 17, 2, 3, 5, 163, 3, 2, 2, 11, 5, 2, 31, 31, 2, 2, 3, 23, 2, 41, 3, 2, 13, 47, 2, 5, 19, 7, 11, 2, 191, 2, 3, 19, 2, 15073, 3, 2, 29, 5, 2, 41, 109, 2, 11, 2, 31, 59, 3, 2, 19, 2, 11, 53, 2, 1019, 137 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Let D be a fundamental discriminant (only the case where D is fundamental is considered because {Kronecker(D,k)} forms a primitive real Dirichlet character with period |D| if and only if D is fundamental), it seems that Sum_{primes q <= p} Kronecker(D,p) <= 0 occurs much more often than its opposite does. This can be seen as a variation of the well-known "Chebyshev's bias". Sequence gives the least prime that violates the inequality when D runs through all negative discriminants.

For any D, the primes p such that Kronecker(D,p) = 1 has asymptotic density 1/2 in all the primes, so a(n) should be > 0 for all n.

Actually, for most n, a(n) is relatively small compared with A003657(n). There are only 127 n's in [1, 3043] (there are 3043 terms in A003657 below 10000) such that a(n) > A003657(n). The largest terms among the 127 corresponding terms are a(1) = 608981813029 (with A003657(1) = 3), a(1955) = 24996194023 (with A003657(1955) = 6240) and a(847) = 1694759239 (with A003657(847) = 2787).

LINKS

Jianing Song, Table of n, a(n) for n = 1..3043

Wikipedia, Chebyshev's bias

EXAMPLE

Let D = -A003657(18) = -52, j(k) = Sum_{primes q <= prime(k)} Kronecker(D,q).

For k = 1, Kronecker(-52,2) = 0, so j(1) = 0;

For k = 2, Kronecker(-52,3) = -1, so j(2) = -1;

For k = 3, Kronecker(-52,5) = -1, so j(3) = -2;

For k = 4, Kronecker(-52,7) = +1, so j(4) = -1;

For k = 5, Kronecker(-52,11) = +1, so j(5) = 0;

For k = 6, Kronecker(-52,13) = 0, so j(6) = 0;

For k = 7, Kronecker(-52,17) = +1, so j(7) = 1.

The first time for j > 0 occurs at the prime 17, so a(18) = 17.

PROG

(PARI) b(n) = my(i=0); forprime(p=2, oo, i+=kronecker(n, p); if(i>0, return(p)))

print1(608981813029, ", "); for(n=4, 300, if(isfundamental(-n), print1(b(-n), ", ")))

CROSSREFS

Cf. A003657, A306499 (the positive discriminants case).

The indices of primes are given in A306503.

Sequence in context: A239921 A250867 A104303 * A306891 A297006 A230559

Adjacent sequences:  A306497 A306498 A306499 * A306501 A306502 A306503

KEYWORD

nonn

AUTHOR

Jianing Song, Feb 19 2019

STATUS

approved

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Last modified February 22 15:05 EST 2020. Contains 332137 sequences. (Running on oeis4.)