

A051025


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).


11



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

1,1


COMMENTS

This is a companion sequence to A051024.
Starting from a(27556)=9103362505801 the sequence includes the 8th signchanging zone predicted by C. Bays et al. The sequence with the first 8 signchanging zones contains 418933 terms (see afile) with a(418933)=9543313015309 as its last term.  Sergei D. Shchebetov, Oct 06 2017
We also discovered the 9th signchanging 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 signchanging 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


LINKS

Andrey S. Shchebetov and Sergei D. Shchebetov, Table of n, a(n) for n = 1..100000
A. Alahmadi, M. Planat, P. Solé, Chebyshev's bias and generalized Riemann hypothesis, HAL Id: hal00650320; Journal of Algebra, Number Theory: Advances and Applications, 2013, 8 (12), pp.4155.
C. Bays and R. H. Hudson, Numerical and graphical description of all axis crossing regions for moduli 4 and 8 which occur before 10^12, International Journal of Mathematics and Mathematical Sciences, vol. 2, no. 1, pp. 111119, 1979.
C. Bays, K. Ford, R. H. Hudson and M. Rubinstein, Zeros of Dirichlet Lfunctions near the real axis and Chebyshev's bias, J. Number Theory 87 (2001), pp.5476.
M. Deléglise, P. Dusart, X. Roblot, Counting Primes in Residue Classes, Mathematics of Computation, American Mathematical Society, 2004, 73 (247), pp.15651575.
A. Granville, G. Martin, Prime Number Races, Amer. Math. Monthly 113 (2006), no. 1, 133.
M. Rubinstein, P. Sarnak, Chebyshev’s bias, Experimental Mathematics, Volume 3, Issue 3, 1994, Pages 173197.
Sergei D. Shchebetov, First 418933 terms (zipped file)
Eric Weisstein's World of Mathematics, Prime Quadratic Effect.


MATHEMATICA

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 *)


CROSSREFS

Cf. A007350, A007351, A038691, A051024, A066520, A096628, A096447, A096448, A199547.
Cf. A156749 Sequence showing Chebyshev bias in prime races (mod 4).  Daniel Forgues, Mar 26 2009
Sequence in context: A235751 A235814 A199547 * A048921 A269115 A249496
Adjacent sequences: A051022 A051023 A051024 * A051026 A051027 A051028


KEYWORD

nonn


AUTHOR

Eric W. Weisstein


EXTENSIONS

Edited by Dean Hickerson, Mar 10 2002


STATUS

approved



