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URL: https://oeis.org/A392294

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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).
2
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
KEYWORD
nonn
AUTHOR
Jianing Song, Jan 06 2026
STATUS
approved