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A007350
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Where prime race 4n-1 vs. 4n+1 changes leader.
(Formerly M3182)
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14
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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;
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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Table of n, a(n) for n=1..33.
A. Granville and G. Martin, Prime number races
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MATHEMATICA
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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 (* From 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 *)
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CROSSREFS
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Cf. A007351, A038691.
Cf. A156749 Sequence showing Chebyshev bias in prime races (mod 4). [From Daniel Forgues, Mar 26 2009]
Sequence in context: A171364 A115475 A225835 * A003839 A175875 A030463
Adjacent sequences: A007347 A007348 A007349 * A007351 A007352 A007353
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v
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STATUS
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approved
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