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%I #66 Feb 03 2018 12:52:44
%S 2946,50378,50380,50382,50392,50414,50418,50420,50422,50424,50426,
%T 50428,50430,50436,50438,50446,50448,50450,50822,50832,50834,50842,
%U 50844,50852,50854,50856,50858,50862,50864,50866,50872,50892,50902
%N Values of n for which pi_{4,3}(p_n) - pi_{4,1}(p_n) = -1, where p_n is the n-th prime and pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).
%C This is a companion sequence to A051025.
%C Starting from a(27556) = 316064952540 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) = 330797040308 as its last term. - _Sergei D. Shchebetov_, Oct 06 2017
%C We also discovered the 9th sign-changing zone, which starts from 2083576475506, ends with 2083615410040, 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
%C We also discovered the 10th sign-changing zone, which starts from 21576098946648, ends with 22056324317296, 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
%H Sergei D. Shchebetov, <a href="/A051024/b051024.txt">Table of n, a(n) for n = 1..100000</a>
%H A. Alahmadi, M. Planat, P. Solé, <a href="https://hal.archives-ouvertes.fr/hal-00650320">Chebyshev's bias and generalized Riemann hypothesis</a>, HAL Id: hal-00650320; Journal of Algebra, Number Theory: Advances and Applications, 2013, 8 (1-2), pp.41-55.
%H C. Bays and R. H. Hudson, <a href="http://dx.doi.org/10.1155/S0161171279000119">Numerical and graphical description of all axis crossing regions for moduli 4 and 8 which occur before 10^12</a>, International Journal of Mathematics and Mathematical Sciences, vol. 2, no. 1, pp. 111-119, 1979.
%H C. Bays, K. Ford, R. H. Hudson and M. Rubinstein, <a href="https://doi.org/10.1006/jnth.2000.2601">Zeros of Dirichlet L-functions near the real axis and Chebyshev's bias</a>, J. Number Theory 87 (2001), pp.54-76.
%H M. Deléglise, P. Dusart, X. Roblot, <a href="http://dx.doi.org/10.1090/S0025-5718-04-01649-7">Counting Primes in Residue Classes</a>, Mathematics of Computation, American Mathematical Society, 2004, 73 (247), pp.1565-1575.
%H A. Granville, G. Martin, <a href="https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/granville1.pdf">Prime Number Races</a>, Amer. Math. Monthly 113 (2006), no. 1, 1-33.
%H M. Rubinstein, P. Sarnak, <a href="https://projecteuclid.org/euclid.em/1048515870">Chebyshev’s bias</a>, Experimental Mathematics, Volume 3, Issue 3, 1994, Pages 173-197.
%H Sergei D. Shchebetov, <a href="https://oeis.org/A051024/a051024-418933.zip">First 418933 terms (zipped file)</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeQuadraticEffect.html">Prime Quadratic Effect.</a>
%t For[i=2; d=0, True, i++, d+=Mod[Prime[i], 4]-2; If[d==-1, Print[i]]]
%t (* Second program: *)
%t Position[Accumulate@ Array[Mod[Prime@ #, 4] - 2 &, 51000], -1][[All, 1]] (* _Michael De Vlieger_, Dec 30 2017 *)
%Y Cf. A007350, A007351, A038691, A051024, A051025, A066520, A096628, A096447, A096448, A199547.
%Y Cf. A156749 (Sequence showing Chebyshev bias in prime races (mod 4)). - _Daniel Forgues_, Mar 26 2009
%K nonn
%O 1,1
%A _Eric W. Weisstein_
%E Edited by _Dean Hickerson_, Mar 05 2002
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