1,1

In general, assuming the strong form of the Riemann Hypothesis, if 0 < a, b < k are integers, gcd(a, k) = gcd(b, k) = 1, a is a quadratic residue and b is a quadratic nonresidue mod k, then Pi(k,b)(n) > Pi(k,a)(n) occurs more often than not. Pi(a,b)(x) denotes the number of primes in the arithmetic progression a*k + b less than or equal to x. This phenomenon is called "Chebyshev's bias". (See Wikipedia link and especially the links in A007350.)

Jianing Song, Table of n, a(n) for n = 1..85027 (a(21049..85027) are terms in the second sign-changing zone of Pi_{3,2}-Pi_{3,1}; see A297006 for more details)

Andrew Granville and Greg Martin, Prime number races, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.

Wikipedia, Chebyshev's bias

(PARI) my(P=608981813000, i=-1); forprime(p=P, P+1e5, i+=kronecker(-3, p); if(i==0, print1(p, ", ")))

Cf. A306891.

Cf. prime indices of ... in A321856: -1 (A297006), 0 (this sequence), 1 (A098044 = prevprime(A096449)), 2 (prevprime(A096452)), 3 (prevprime(A096453)).

Sequence in context: A239921 A250867 A104303 * A306500 A306891 A297006

Adjacent sequences: A392291 A392292 A392293 * A392295 A392296 A392297

Jianing Song, Jan 06 2026

approved