A007350 Where the prime race 4k-1 vs. 4k+1 changes leader.

3, 26861, 26879, 616841, 617039, 617269, 617471, 617521, 617587, 617689, 617723, 622813, 623387, 623401, 623851, 623933, 624031, 624097, 624191, 624241, 624259, 626929, 626963, 627353, 627391, 627449, 627511, 627733, 627919, 628013, 628427, 628937, 629371

Offset: 1

Author

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

Keywords

• nonn
• easy

Comments

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

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

References

• 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.
• 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.
• 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 (terms 1..125 from Vincenzo Librandi, terms 126..301 from Robert G. Wilson v)
• 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.
• Andrey Feuerverger and Greg Martin, Biases in the Shanks-Renyi prime number race Experiment. Math. 9 (2000), no. 4, 535-570.
• Kevin Ford and Sergei Konyagin, Chebyshev's conjecture and the prime number race, 2002.
• Kevin Ford and Sergei Konyagin, The prime number race and zeros of L-functions off the critical line, Duke Math. J., Volume 113, Number 2 (2002), 313-330.
• Kevin Ford and Sergei Konyagin, The prime number race and zeros of L-functions off the critical line. II, Proceedings of the Session in Analytic Number Theory and Diophantine Equations, 40 pp., Bonner Math. Schriften, 360, 2003.
• A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.
• Andrew Granville and Greg 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]
• Jerzy Kaczorowski, A contribution to the Shanks-Renyi race problem, Quart. J. Math. Oxford Ser. (2) 44 (1993), no. 176, 451-458.
• Jerzy Kaczorowski, On the Shanks-Renyi race problem mod 5. J. Number Theory 50 (1995), no. 1, 106-118.
• Greg Martin, Asymmetries in the Shanks-Renyi prime number race, arXiv:math/0010086 [math.NT], 2000; Number theory for the millennium, II (Urbana, IL, 2000), 403-415, A K Peters, Natick, MA, 2002.
• J.-C. Puchta, On large oscillations of the remainder of the prime number theorems Acta Math. Hungar. 87 (2000), no. 3, 213-227.
• M. Rubinstein and P. Sarnak, Chebyshev's bias, Exper. Math., 3 (1994), 173-197.
• Andrey S. Shchebetov and Sergei D. Shchebetov, First 194367 terms (zipped file).
• Robert G. Wilson v, Letter to N. J. A. Sloane, Aug. 1993.
• G. M. Ziegler, The great prime number record races, Notices Amer. Math. Soc. 51 (2004), no. 4, 414-416.

Crossrefs

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).

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

Programs

Mathematica

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 *)
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 *)