A297448 Primes p for which pi_{8,5}(p) - pi_{8,1}(p) = -1, where pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).

588067889, 588068009, 588068093, 588068309, 588068333, 588068477, 588068717, 588069137, 588069169, 588069409, 588069529, 588069809, 588070897, 588070949, 588071009, 588071101, 588071401, 588071573, 588071597, 588079253

Offset: 1

Author

Andrey S. Shchebetov and Sergei D. Shchebetov, Dec 30 2017

Keywords

• nonn

Comments

This is a companion sequence to A297447. The first two sign-changing zones were discovered by Bays and Hudson back in 1979. We discovered four additional zones starting from a(22794) = 5267226902633. The full sequence with all 6 zones checked up to 5*10^14 contains 664175 terms (see a-file) with a(664175) = 194318969449909 as its last term.

This sequence was checked up to 10^15 and the 7th sign-changing zone starting from a(664176) = 930525161507057 and ending with a(850232)= 932080335660277 was found. - Andrey S. Shchebetov and Sergei D. Shchebetov, Jul 28 2018

The y-coordinate of the terms in this sequence on the Cartesian grid defined in A379643 is -1. - Ya-Ping Lu, Jan 07 2025

Links

• Sergei D. Shchebetov, Table of n, a(n) for n = 1..100000
• C. Bays and R. H. Hudson, Numerical and graphical description of all axis crossing regions for moduli 4 and 8 which occur before 10^12, International Journal of Mathematics and Mathematical Sciences, vol. 2, no. 1, pp. 111-119, 1979.
• C. Bays, K. Ford, R. H. Hudson and M. Rubinstein, Zeros of Dirichlet L-functions near the real axis and Chebyshev's bias, J. Number Theory 87 (2001), pp.54-76.
• M. Deléglise, P. Dusart, and X. Roblot, Counting Primes in Residue Classes, Mathematics of Computation, American Mathematical Society, 2004, 73 (247), pp.1565-1575.
• A. Granville and G. Martin, Prime Number Races, Amer. Math. Monthly 113 (2006), no. 1, 1-33.
• M. Rubinstein and P. Sarnak, Chebyshev’s bias, Experimental Mathematics, Volume 3, Issue 3, 1994, Pages 173-197.
• Eric Weisstein's World of Mathematics, Prime Quadratic Effect.

Crossrefs

Cf. A297447, A379643, A379731.

Programs

Python

from sympy import nextprime; p, r1, r5 = 1, 0, 0
while p <= 588079253:
    p = nextprime(p); r = p%8
    if r == 1: r1 += 1
    elif r == 5: r5 += 1
if r in {1, 5} and r1 == r5 + 1: print(p, end = ', ')  # Ya-Ping Lu, Jan 07 2025