A392294 Primes p such that Pi_{3,1}(p) = Pi_{3,2}(p), where Pi_{m,n}(p) denotes the number of primes q <= p with q == n (mod m).

608981813017, 608981813123, 608981813303, 608981813501, 608981813569, 608981813677, 608981813807, 608981813833, 608981813851, 608981814043, 608981814131, 608981818987, 608981819339, 608981819393, 608981820911, 608981820917, 608981826853, 608981826941, 608981826991

Offset

1,1

Comments

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.)

Links

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

Prog

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

Crossrefs

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

Keyword

nonn

Author

Jianing Song, Jan 06 2026

Status

approved