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.

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

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