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A007351 Where prime race 4m-1 vs. 4m+1 is tied.
(Formerly M1507)
23
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 (list; graph; refs; listen; history; text; internal format)
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, 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, 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, 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)/a>
Eric Weisstein's World of Mathematics, Prime Quadratic Effect.
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. 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
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Corrected and extended by Enoch Haga, Feb 24 2004
STATUS
approved

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Last modified March 19 07:04 EDT 2024. Contains 370953 sequences. (Running on oeis4.)