OFFSET
1,1
COMMENTS
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
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: hal-00650320; Journal of Algebra, Number Theory: Advances and Applications, 2013, 8 (1-2), pp.41-55.
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. 111-119, 1979.
C. Bays, K. Ford, R. H. Hudson and M. Rubinstein, Zeros of Dirichlet L-functions near the real axis and Chebyshev's bias, J. Number Theory 87 (2001), pp.54-76.
M. Deléglise, P. Dusart, X. Roblot, Counting Primes in Residue Classes, Mathematics of Computation, American Mathematical Society, 2004, 73 (247), pp.1565-1575.
A. Granville, G. Martin, Prime Number Races, Amer. Math. Monthly 113 (2006), no. 1, 1-33.
M. Rubinstein, P. Sarnak, Chebyshev’s bias, Experimental Mathematics, Volume 3, Issue 3, 1994, Pages 173-197.
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. A156749 Sequence showing Chebyshev bias in prime races (mod 4). - Daniel Forgues, Mar 26 2009
KEYWORD
nonn
AUTHOR
EXTENSIONS
Edited by Dean Hickerson, Mar 10 2002
STATUS
approved