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A051025
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Primes p for which pi_{4,3}(p) - pi_{4,1}(p) = -1, where pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).
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11
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26861, 616841, 616849, 616877, 617011, 617269, 617327, 617339, 617359, 617369, 617401, 617429, 617453, 617521, 617537, 617689, 617699, 617717, 622813, 622987, 623003, 623107, 623209, 623299, 623321, 623341, 623353, 623401, 623423, 623437
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OFFSET
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1,1
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COMMENTS
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This is a companion sequence to A051024.
Starting from a(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 418933 terms (see a-file) with a(418933)=9543313015309 as its last term. - Sergei D. Shchebetov, Oct 06 2017
We also discovered the 9th sign-changing zone, which starts from 64083080712569, ends with 64084318523021, and has 13370 terms with pi_{4,3}(p) - pi_{4,1}(p) = -1. This zone is considerably lower than predicted by M. Deléglise et al. in 2004. - Andrey S. Shchebetov and Sergei D. Shchebetov, Dec 30 2017
We also discovered the 10th sign-changing zone, which starts from 715725135905981, ends with 732156384107921, and has 481194 terms with pi_{4,3}(p) - pi_{4,1}(p) = -1. This zone is considerably lower than predicted by M. Deléglise et al. in 2004. - Andrey S. Shchebetov and Sergei D. Shchebetov, Jan 28 2018
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LINKS
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M. Rubinstein, P. Sarnak, Chebyshev’s bias, Experimental Mathematics, Volume 3, Issue 3, 1994, Pages 173-197.
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MATHEMATICA
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For[i=2; d=0, True, i++, d+=Mod[p=Prime[i], 4]-2; If[d==-1, Print[p]]]
(* Second program: *)
Prime@ Position[Accumulate@ Array[Mod[Prime@ #, 4] - 2 &, 51000], -1][[All, 1]] (* Michael De Vlieger, Dec 30 2017 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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