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A007350
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Where the prime race 4k-1 vs. 4k+1 changes leader.
(Formerly M3182)
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38
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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;
graph;
refs;
listen;
history;
text;
internal format)
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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 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
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REFERENCES
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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).
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LINKS
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Andrew Granville and Greg Martin, Prime number races, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.
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.
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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 (* 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. A007350, A007351, A007352, A007353, A007354, A297406, A297407, A297408, A297410, A297411, A038691, A051024, A066520, A096628, A096447, A096448, A199547, A038698 (another way to watch this race).
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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