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A007350 Where prime race 4n-1 vs. 4n+1 changes leader.
(Formerly M3182)
14
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The following references include some on the "prime race" question that are not necessarily related to this sequence. - N. J. A. Sloane, May 22 2006

REFERENCES

Feuerverger, Andrey; Martin, Greg; Biases in the Shanks-Renyi prime number race. Experiment. Math. 9 (2000), no. 4, 535-570.

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.

Ford, Kevin; Konyagin, Sergei; 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.

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.

A. Granville and G. Martin, Prime number races, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.

Kaczorowski, Jerzy; A contribution to the Shanks-Renyi race problem. Quart. J. Math. Oxford Ser. (2) 44 (1993), no. 176, 451-458.

Kaczorowski, Jerzy; On the Shanks-Renyi race problem mod 5. J. Number Theory 50 (1995), no. 1, 106-118.

Martin, Greg; Asymmetries in the Shanks-Renyi prime number race. Number theory for the millennium, II (Urbana, IL, 2000), 403-415, A K Peters, Natick, MA, 2002.

Puchta, J.-C.; 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.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Ziegler, G. M. The great prime number record races. Notices Amer. Math. Soc. 51 (2004), no. 4, 414-416.

LINKS

Vincenzo Librandi and Robert G. Wilson v, Table of n, a(n) for n = 1..301 (first 125 terms from Vincenzo Librandi)

A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004

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-Fran├žois 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. A007351, A038691.

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

Sequence in context: A171364 A115475 A225835 * A003839 A175875 A030463

Adjacent sequences:  A007347 A007348 A007349 * A007351 A007352 A007353

KEYWORD

nonn,easy

AUTHOR

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

STATUS

approved

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Last modified December 22 00:41 EST 2014. Contains 252326 sequences.