A007351 Where prime race 4m-1 vs. 4m+1 is tied. (Formerly M1507)

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

1,1

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

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 *)

Prog

(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

Crossrefs

Cf. A007350, A038691.

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

Sequence in context: A346134 A081763 A013918 * A300692 A076076 A136194

Adjacent sequences: A007348 A007349 A007350 * A007352 A007353 A007354

Keyword

nonn,easy

Author

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

Extensions

Corrected and extended by Enoch Haga, Feb 24 2004

Status

approved