

A007350


Where prime race 4n1 vs. 4n+1 changes leader.
(Formerly M3182)


30



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
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OFFSET

1,1


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 signchanging zone predicted by C. Bays et al. The sequence with the first 8 signchanging zones contains 194367 terms (see afile) 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), 6791.
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, 197240.
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 301 terms from Vincenzo Librandi and Robert G. Wilson)
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. 111119, 1979.
C. Bays, K. Ford, R. H. Hudson and M. Rubinstein, Zeros of Dirichlet Lfunctions near the real axis and Chebyshev's bias, J. Number Theory 87 (2001), pp.5476.
M. Deléglise, P. Dusart, X. Roblot, Counting Primes in Residue Classes, Mathematics of Computation, American Mathematical Society, 2004, 73 (247), pp.15651575.
Andrey Feuerverger, Greg Martin, Biases in the ShanksRenyi prime number race Experiment. Math. 9 (2000), no. 4, 535570.
Kevin Ford, Sergei Konyagin, Chebyshev's conjecture and the prime number race, 2002.
Kevin Ford, Sergei Konyagin, The prime number race and zeros of Lfunctions off the critical line, Duke Math. J., Volume 113, Number 2 (2002), 313330.
Kevin Ford, Sergei Konyagin, The prime number race and zeros of Lfunctions 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), 133.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697712. [Annotated scanned copy]
Jerzy Kaczorowski, A contribution to the ShanksRenyi race problem, Quart. J. Math. Oxford Ser. (2) 44 (1993), no. 176, 451458.
Jerzy Kaczorowski, On the ShanksRenyi race problem mod 5. J. Number Theory 50 (1995), no. 1, 106118.
Greg Martin, Asymmetries in the ShanksRenyi prime number race, arXiv:math/0010086 [math.NT], 2000; Number theory for the millennium, II (Urbana, IL, 2000), 403415, 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, 213227.
M. Rubinstein and P. Sarnak, Chebyshev's bias, Exper. Math., 3 (1994), 173197.
Andrey S. Shchebetov and Sergei D. Shchebetov, First 194367 terms (zipped file)/a>
R. 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, 414416.


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[[k1, 2]] < t[[k1, 3]] && t[[k, 2]] == t[[k, 3]] && t[[k+1, 2]] > t[[k+1, 3]]  t[[k1, 2]] > t[[k1, 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 (* JeanFrançois 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 *)


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
Sequence in context: A171364 A115475 A225835 * A003839 A297023 A175875
Adjacent sequences: A007347 A007348 A007349 * A007351 A007352 A007353


KEYWORD

nonn,easy,changed


AUTHOR

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


STATUS

approved



