cp's OEIS Frontend

A051025 Primes p for which pi_{4,3}(p) - pi_{4,1}(p) = -1, where pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).

%I A051025 #59 May 27 2025 08:37:20
%S A051025 26861,616841,616849,616877,617011,617269,617327,617339,617359,617369,
%T A051025 617401,617429,617453,617521,617537,617689,617699,617717,622813,
%U A051025 622987,623003,623107,623209,623299,623321,623341,623353,623401,623423,623437
%N A051025 Primes p for which pi_{4,3}(p) - pi_{4,1}(p) = -1, where pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).
%C A051025 This is a companion sequence to A051024.
%C A051025 Starting from a(27556)=9103362505801 the sequence includes the 8th sign-changing zone predicted by C. Bays et al. The sequence with the first 8 sign-changing zones contains 418933 terms (see a-file) with a(418933)=9543313015309 as its last term. - _Sergei D. Shchebetov_, Oct 06 2017
%C A051025 We also discovered the 9th sign-changing zone, which starts from 64083080712569, ends with 64084318523021, and has 13370 terms with pi_{4,3}(p) - pi_{4,1}(p) = -1. This zone is considerably lower than predicted by M. Deléglise et al. in 2004. - Andrey S. Shchebetov and _Sergei D. Shchebetov_, Dec 30 2017
%C A051025 We also discovered the 10th sign-changing zone, which starts from 715725135905981, ends with 732156384107921, and has 481194 terms with pi_{4,3}(p) - pi_{4,1}(p) = -1. This zone is considerably lower than predicted by M. Deléglise et al. in 2004. - Andrey S. Shchebetov and _Sergei D. Shchebetov_, Jan 28 2018
%H A051025 Andrey S. Shchebetov and Sergei D. Shchebetov, <a href="/A051025/b051025.txt">Table of n, a(n) for n = 1..100000</a>
%H A051025 A. Alahmadi, M. Planat, and P. Solé, <a href="https://hal.archives-ouvertes.fr/hal-00650320">Chebyshev's bias and generalized Riemann hypothesis</a>, HAL Id: hal-00650320; Journal of Algebra, Number Theory: Advances and Applications, 2013, 8 (1-2), pp.41-55.
%H A051025 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 A051025 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 A051025 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 A051025 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 A051025 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 A051025 Sergei D. Shchebetov, <a href="https://oeis.org/A051025/a051025-418933.zip">First 418933 terms (zipped file)</a>
%H A051025 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeQuadraticEffect.html">Prime Quadratic Effect.</a>
%t A051025 For[i=2; d=0, True, i++, d+=Mod[p=Prime[i], 4]-2; If[d==-1, Print[p]]]
%t A051025 (* Second program: *)
%t A051025 Prime@ Position[Accumulate@ Array[Mod[Prime@ #, 4] - 2 &, 51000], -1][[All, 1]] (* _Michael De Vlieger_, Dec 30 2017 *)
%o A051025 (Python)
%o A051025 from sympy import nextprime; a, p = 0, 2
%o A051025 while p < 623803:
%o A051025     p=nextprime(p); a += p%4-2
%o A051025     if a == -1: print(p, end = ', ')  # _Ya-Ping Lu_, Jan 18 2025
%Y A051025 Cf. A007350, A007351, A038691, A051024, A066520, A096628, A096447, A096448, A199547.
%Y A051025 Cf. A156749 Sequence showing Chebyshev bias in prime races (mod 4). - _Daniel Forgues_, Mar 26 2009
%K A051025 nonn
%O A051025 1,1
%A A051025 _Eric W. Weisstein_
%E A051025 Edited by _Dean Hickerson_, Mar 10 2002