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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).

%I A297448 #25 May 27 2025 10:09:18
%S A297448 588067889,588068009,588068093,588068309,588068333,588068477,
%T A297448 588068717,588069137,588069169,588069409,588069529,588069809,
%U A297448 588070897,588070949,588071009,588071101,588071401,588071573,588071597,588079253
%N 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).
%C A297448 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.
%C A297448 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
%C A297448 The y-coordinate of the terms in this sequence on the Cartesian grid defined in A379643 is -1. - _Ya-Ping Lu_, Jan 07 2025
%H A297448 Sergei D. Shchebetov, <a href="/A297448/b297448.txt">Table of n, a(n) for n = 1..100000</a>
%H A297448 C. Bays and R. H. Hudson, <a href="http://dx.doi.org/10.1155/S0161171279000119">Numerical and graphical description of all axis crossing regions for moduli 4 and 8 which occur before 10^12</a>, International Journal of Mathematics and Mathematical Sciences, vol. 2, no. 1, pp. 111-119, 1979.
%H A297448 C. Bays, K. Ford, R. H. Hudson and M. Rubinstein, <a href="https://doi.org/10.1006/jnth.2000.2601">Zeros of Dirichlet L-functions near the real axis and Chebyshev's bias</a>, J. Number Theory 87 (2001), pp.54-76.
%H A297448 M. Deléglise, P. Dusart, and X. Roblot, <a href="http://dx.doi.org/10.1090/S0025-5718-04-01649-7">Counting Primes in Residue Classes</a>, Mathematics of Computation, American Mathematical Society, 2004, 73 (247), pp.1565-1575.
%H A297448 A. Granville and G. Martin, <a href="https://web.archive.org/web/20240529054811/https://maa.org/sites/default/files/pdf/upload_library/22/Ford/granville1.pdf">Prime Number Races</a>, Amer. Math. Monthly 113 (2006), no. 1, 1-33.
%H A297448 M. Rubinstein and P. Sarnak, <a href="https://projecteuclid.org/euclid.em/1048515870">Chebyshev’s bias</a>, Experimental Mathematics, Volume 3, Issue 3, 1994, Pages 173-197.
%H A297448 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeQuadraticEffect.html">Prime Quadratic Effect.</a>
%o A297448 (Python)
%o A297448 from sympy import nextprime; p, r1, r5 = 1, 0, 0
%o A297448 while p <= 588079253:
%o A297448     p = nextprime(p); r = p%8
%o A297448     if r == 1: r1 += 1
%o A297448     elif r == 5: r5 += 1
%o A297448 if r in {1, 5} and r1 == r5 + 1: print(p, end = ', ')  # _Ya-Ping Lu_, Jan 07 2025
%Y A297448 Cf. A297447, A379643, A379731.
%K A297448 nonn
%O A297448 1,1
%A A297448 Andrey S. Shchebetov and _Sergei D. Shchebetov_, Dec 30 2017