A392297 Primes p such that Pi_{5,2}(p) + Pi_{5,3}(p) - Pi_{5,1}(p) - Pi_{5,4}(p) = 1, where Pi_{m,n}(p) denotes the number of primes q <= p with q == n (mod m).

2, 2082927019, 2082927191, 2082928163, 2082928193, 2082941183, 2082941209, 2082941297, 2082941323, 2082941389, 2082941489, 2082941533, 2082941569, 2082944527, 2082944557, 2082944593, 2082944603, 2082944623, 2082944713, 2082944723, 2082945089, 2082945197, 2082945239, 2082945569

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..981 (all terms up to the first region where Pi_{5,2}+Pi_{5,3}-Pi_{5,1}-Pi_{5,4} is negative)

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

Wikipedia, Chebyshev's bias

Prog

(PARI) my(i=0); forprime(p=2, 2083000000, i+=kronecker(5, p); if(i==-1, print1(p, ", ")))

Crossrefs

Cf. prime indices of -1, 0, 1, 2, 3 in A321857: A392295, A392296, this sequence, A392298, A392299.

Sequence in context: A249482 A214600 A325175 * A306499 A051241 A094486

Adjacent sequences: A392294 A392295 A392296 * A392298 A392299 A392300

Keyword

nonn

Author

Jianing Song, Jan 08 2026

Status

approved