A051024 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).

2946, 50378, 50380, 50382, 50392, 50414, 50418, 50420, 50422, 50424, 50426, 50428, 50430, 50436, 50438, 50446, 50448, 50450, 50822, 50832, 50834, 50842, 50844, 50852, 50854, 50856, 50858, 50862, 50864, 50866, 50872, 50892, 50902

Offset

1,1

Comments

This is a companion sequence to A051025.

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

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

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

Links

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[Prime[i], 4]-2; If[d==-1, Print[i]]]
(* Second program: *)
Position[Accumulate@ Array[Mod[Prime@ #, 4] - 2 &, 51000], -1][[All, 1]] (* Michael De Vlieger, Dec 30 2017 *)

Crossrefs

Cf. A007350, A007351, A038691, A051024, A051025, A066520, A096628, A096447, A096448, A199547.

Cf. A156749 (Sequence showing Chebyshev bias in prime races (mod 4)). - Daniel Forgues, Mar 26 2009

Sequence in context: A231313 A259999 A096628 * A177087 A125907 A025514

Adjacent sequences: A051021 A051022 A051023 * A051025 A051026 A051027

Keyword

nonn

Author

Eric W. Weisstein

Extensions

Edited by Dean Hickerson, Mar 05 2002

Status

approved