%I M3182 #94 Oct 30 2023 00:47:57
%S 3,26861,26879,616841,617039,617269,617471,617521,617587,617689,
%T 617723,622813,623387,623401,623851,623933,624031,624097,624191,
%U 624241,624259,626929,626963,627353,627391,627449,627511,627733,627919,628013,628427,628937,629371
%N Where the prime race 4k-1 vs. 4k+1 changes leader.
%C The following references include some on the "prime race" question that are not necessarily related to this particular sequence. - _N. J. A. Sloane_, May 22 2006
%C Starting from a(12502) = A051025(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 194367 terms (see a-file) with a(194367) = 9543313015387 as its last term. - _Sergei D. Shchebetov_, Oct 13 2017
%D Ford, Kevin; Konyagin, Sergei; Chebyshev's conjecture and the prime number race. IV International Conference "Modern Problems of Number Theory and its Applications": Current Problems, Part II (Russian) (Tula, 2001), 67-91.
%D Granville, Andrew; Martin, Greg; Prime number races. (Spanish) With appendices by Giuliana Davidoff and Michael Guy. Gac. R. Soc. Mat. Esp. 8 (2005), no. 1, 197-240.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Andrey S. Shchebetov and Sergei D. Shchebetov, <a href="/A007350/b007350.txt">Table of n, a(n) for n = 1..100000</a> (terms 1..125 from Vincenzo Librandi, terms 126..301 from Robert G. Wilson v)
%H C. Bays and R. H. Hudson, <a href="https://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 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 M. Deléglise, P. Dusart and X. Roblot, <a href="https://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 Andrey Feuerverger and Greg Martin, <a href="https://www.emis.de/journals/EM/restricted/9/9.4/feuer.ps">Biases in the Shanks-Renyi prime number race</a> Experiment. Math. 9 (2000), no. 4, 535-570.
%H Kevin Ford and Sergei Konyagin, <a href="https://pdfs.semanticscholar.org/e29b/80d3ca8a3cec35c30a69f5346e133c371db3.pdf">Chebyshev's conjecture and the prime number race</a>, 2002.
%H Kevin Ford and Sergei Konyagin, <a href="https://pdfs.semanticscholar.org/aa47/b18a56bec41c12b5e4c3faed20e76fd91b0e.pdf">The prime number race and zeros of L-functions off the critical line</a>, Duke Math. J., Volume 113, Number 2 (2002), 313-330.
%H Kevin Ford and Sergei Konyagin, <a href="https://faculty.math.illinois.edu/~ford/wwwpapers/barriers2.pdf">The prime number race and zeros of L-functions off the critical line. II</a>, Proceedings of the Session in Analytic Number Theory and Diophantine Equations, 40 pp., Bonner Math. Schriften, 360, 2003.
%H A. Granville and G. Martin, <a href="https://arxiv.org/abs/math/0408319">Prime number races</a>, arXiv:math/0408319 [math.NT], 2004.
%H Andrew Granville and Greg Martin, <a href="http://www.jstor.org/stable/27641834">Prime number races</a>, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.
%H 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 Jerzy Kaczorowski, <a href="http://www.math.ubc.ca/~gerg/teaching/592-Fall2018/papers/1993.Kaczorowski.pdf">A contribution to the Shanks-Renyi race problem</a>, Quart. J. Math. Oxford Ser. (2) 44 (1993), no. 176, 451-458.
%H Jerzy Kaczorowski, <a href="https://doi.org/10.1006/jnth.1995.1006">On the Shanks-Renyi race problem mod 5</a>. J. Number Theory 50 (1995), no. 1, 106-118.
%H Greg Martin, <a href="https://arxiv.org/abs/math/0010086">Asymmetries in the Shanks-Renyi prime number race</a>, arXiv:math/0010086 [math.NT], 2000; Number theory for the millennium, II (Urbana, IL, 2000), 403-415, A K Peters, Natick, MA, 2002.
%H J.-C. Puchta, <a href="http://www.math.ubc.ca/~gerg/teaching/592-Fall2018/papers/2000.Puchta.pdf">On large oscillations of the remainder of the prime number theorems</a> Acta Math. Hungar. 87 (2000), no. 3, 213-227.
%H M. Rubinstein and P. Sarnak, <a href="https://projecteuclid.org/euclid.em/1048515870">Chebyshev's bias</a>, Exper. Math., 3 (1994), 173-197.
%H Andrey S. Shchebetov and Sergei D. Shchebetov, <a href="/A007350/a007350-194367.zip">First 194367 terms (zipped file)</a>.
%H Robert G. Wilson v, <a href="/A005596/a005596.pdf">Letter to N. J. A. Sloane, Aug. 1993</a>.
%H G. M. Ziegler, <a href="http://www.ams.org/notices/200404/comm-ziegler.pdf">The great prime number record races</a>, Notices Amer. Math. Soc. 51 (2004), no. 4, 414-416.
%t lim = 10^5; k1 = 0; k3 = 0; t = Table[{p = Prime[k], If[Mod[p, 4] == 1, ++k1, k1], If[Mod[p, 4] == 3, ++k3, k3]}, {k, 2, lim}]; A007350 = {3}; Do[ If[t[[k-1, 2]] < t[[k-1, 3]] && t[[k, 2]] == t[[k, 3]] && t[[k+1, 2]] > t[[k+1, 3]] || t[[k-1, 2]] > t[[k-1, 3]] && t[[k, 2]] == t[[k, 3]] && t[[k+1, 2]] < t[[k+1, 3]], AppendTo[A007350, t[[k+1, 1]]]], {k, 2, Length[t]-1}]; A007350 (* _Jean-François Alcover_, Sep 07 2011 *)
%t lim = 10^5; k1 = 0; k3 = 0; p = 2; t = {}; parity = Mod[p, 4]; Do[p = NextPrime[p]; If[Mod[p, 4] == 1, k1++, k3++]; If[(k1 - k3)*(parity - Mod[p, 4]) > 0, AppendTo[t, p]; parity = Mod[p, 4]], {lim}]; t (* _T. D. Noe_, Sep 07 2011 *)
%Y Cf. A007350, A007351, A007352, A007353, A007354, A297406, A297407, A297408, A297410, A297411, A038691, A051024, A066520, A096628, A096447, A096448, A199547, A038698 (another way to watch this race).
%Y Cf. A156749 [sequence showing Chebyshev bias in prime races (mod 4)]. - _Daniel Forgues_, Mar 26 2009
%K nonn,easy
%O 1,1
%A _N. J. A. Sloane_, _Mira Bernstein_, _Robert G. Wilson v_