cp's OEIS Frontend

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

%I A007351 M1507 #56 Feb 16 2025 08:32:31
%S A007351 2,5,17,41,461,26833,26849,26863,26881,26893,26921,616769,616793,
%T A007351 616829,616843,616871,617027,617257,617363,617387,617411,617447,
%U A007351 617467,617473,617509,617531,617579,617681,617707,617719,618437,618521,618593,618637
%N A007351 Where prime race 4m-1 vs. 4m+1 is tied.
%C A007351 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.
%C A007351 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
%D A007351 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007351 Andrey S. Shchebetov and Sergei D. Shchebetov, <a href="/A007351/b007351.txt">Table of n, a(n) for n = 1..100000</a> (first 1000 terms from T. D. Noe)
%H A007351 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.
%H A007351 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 A007351 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 A007351 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 A007351 A. Granville and G. Martin, <a href="http://www.jstor.org/stable/27641834">Prime number races</a>, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.
%H A007351 R. K. Guy, <a href="/A005165/a005165.pdf">The strong law of small numbers</a>. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
%H A007351 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 A007351 Andrey S. Shchebetov and Sergei D. Shchebetov, <a href="/A007351/a007351-419467.zip">First 419467 terms (zipped file)</a>
%H A007351 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeQuadraticEffect.html">Prime Quadratic Effect</a>.
%H A007351 Robert G. Wilson v, <a href="/A005596/a005596.pdf">Letter to N. J. A. Sloane, Aug. 1993</a>
%t A007351 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 *)
%o A007351 (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
%o A007351 (Python)
%o A007351 from sympy import nextprime; a, p = 0, 2; R = [p]
%o A007351 while p < 618637:
%o A007351     p=nextprime(p); a += p%4-2
%o A007351     if a == 0: R.append(p)
%o A007351 print(*R, sep = ', ')  # _Ya-Ping Lu_, Jan 18 2025
%Y A007351 Cf. A007350, A038691.
%Y A007351 Cf. A156749 Sequence showing Chebyshev bias in prime races (mod 4). [From _Daniel Forgues_, Mar 26 2009]
%K A007351 nonn,easy
%O A007351 1,1
%A A007351 _N. J. A. Sloane_, _Mira Bernstein_, _Robert G. Wilson v_
%E A007351 Corrected and extended by _Enoch Haga_, Feb 24 2004