A007351 Where prime race 4m-1 vs. 4m+1 is tied.

2, 5, 17, 41, 461, 26833, 26849, 26863, 26881, 26893, 26921, 616769, 616793, 616829, 616843, 616871, 617027, 617257, 617363, 617387, 617411, 617447, 617467, 617473, 617509, 617531, 617579, 617681, 617707, 617719, 618437, 618521, 618593, 618637

Offset: 1

Author

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

Keywords

• nonn
• easy

Comments

Primes p such that the number of primes <= p of the form 4m-1 is equal to the number of primes <= p of the form 4m+1.

Starting from a(27410)=9103362505753 the sequence includes the 8th sign-changing zone predicted by C. Bays et al. The sequence with the first 8 sign-changing zones contains 419467 terms (see a-file) with a(419467)=9543313015351 as its last term. - Sergei D. Shchebetov, Oct 15 2017

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Andrey S. Shchebetov and Sergei D. Shchebetov, Table of n, a(n) for n = 1..100000 (first 1000 terms from T. D. Noe)
• A. Alahmadi, M. Planat, and P. Solé, Chebyshev's bias and generalized Riemann hypothesis, HAL Id: hal-00650320.
• 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 (No. 1, 2006), 1-33.
• R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
• M. Rubinstein and P. Sarnak, Chebyshev’s bias, Experimental Mathematics, Volume 3, Issue 3, 1994, Pages 173-197.
• Andrey S. Shchebetov and Sergei D. Shchebetov, First 419467 terms (zipped file)
• Eric Weisstein's World of Mathematics, Prime Quadratic Effect.
• Robert G. Wilson v, Letter to N. J. A. Sloane, Aug. 1993

Crossrefs

Cf. A007350, A038691.

Cf. A156749 Sequence showing Chebyshev bias in prime races (mod 4). [From Daniel Forgues, Mar 26 2009]

Programs

Mathematica

Prime@ Position[Fold[Append[#1, #1[[-1]] + If[Mod[#2, 4] == 3, {1, 0}, {0, 1}]] &, {{0, 0}}, Prime@ Range[2, 10^5]], ?(SameQ @@ # &)][[All, 1]] (* _Michael De Vlieger, May 27 2018 *)

PARI

lista(nn) = {nb = 0; forprime(p=2, nn, m = (p % 4); if (m == 1, nb++, if (m == 3, nb--)); if (!nb, print1(p, ", ")););} \\ Michel Marcus, Oct 05 2017

Python

from sympy import nextprime; a, p = 0, 2; R = [p]
while p < 618637:
    p=nextprime(p); a += p%4-2
    if a == 0: R.append(p)
print(*R, sep = ', ')  # Ya-Ping Lu, Jan 18 2025

Extensions

Corrected and extended by Enoch Haga, Feb 24 2004