A392295 - OEIS

A392295

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).

%I #12 Jan 08 2026 21:51:07

%S 2082927221,2082927367,2082927443,2082927481,2082928013,2082928117,

%T 2082928229,2082940207,2082940229,2082940243,2082940663,2082940669,

%U 2082940723,2082940777,2082940799,2082940829,2082940879,2082940913,2082940943,2082940991,2082941039,2082941131,2082941591,2082943883

%N 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).

%C 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.)

%H Jianing Song, <a href="/A392295/b392295.txt">Table of n, a(n) for n = 1..1090</a> (all terms up to the first region where Pi_{5,2}+Pi_{5,3}-Pi_{5,1}-Pi_{5,4} is negative)

%H Andrew Granville and Greg Martin, <a href="https://www.jstor.org/stable/27641834">Prime number races</a>, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Chebyshev%27s_bias">Chebyshev's bias</a>

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

%Y Cf. prime indices of -1, 0, 1, 2, 3 in A321857: this sequence, A392296, A392297, A392298, A392299.

%K nonn

%O 1,1

%A _Jianing Song_, Jan 08 2026