A297005 Values of n for which pi_{3,2}(p_n) - pi_{3,1}(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).

23338590792, 23338590794, 23338590796, 23338590798, 23338590800, 23338590802, 23338590804, 23338590810, 23338590814, 23338590816, 23338590818, 23338590832, 23338591016, 23338591018, 23338591028, 23338591030, 23338591032, 23338591084, 23338591086, 23338591088, 23338591302, 23338591340, 23338591342, 23338591344, 23338591346, 23338591348, 23338591350, 23338591656, 23338591658, 23338591662

Offset: 1

Author

Andrey S. Shchebetov and Sergei D. Shchebetov, Dec 23 2017

Keywords

• nonn

Comments

This sequence is a companion sequence to A297006. Starting from a(20591)=216415270060 the sequence includes the second sign-changing zone predicted by C. Bays et al. in 2001. The sequence with the first two sign-changing zones up to 10^13 contains 84323 terms with a(84323)=216682882512 as its last term (see b-file). In addition, a(1) = A096630(1).

Links

• Sergei D. Shchebetov, Table of n, a(n) for n = 1..84323
• A. Alahmadi, M. Planat, and P. Solé, Chebyshev's bias and generalized Riemann hypothesis, HAL Id: hal-00650320.
• C. Bays and R. H. Hudson, Details of the first region of integers x with pi_{3,2} (x) < pi_{3,1}(x), Math. Comp. 32 (1978), 571-576.
• 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, and X. Roblot, Counting Primes in Residue Classes, Mathematics of Computation, American Mathematical Society, 2004, 73 (247), pp. 1565-1575.
• A. Granville and G. Martin, Prime Number Races, Amer. Math. Monthly 113 (2006), no. 1, 1-33.
• M. Rubinstein and P. Sarnak, Chebyshev's bias, Experimental Mathematics, Volume 3, Issue 3, 1994, pp. 173-197.
• Eric Weisstein's World of Mathematics, Prime Quadratic Effect.

Crossrefs

Cf. A007352, A096629, A096630, A096449, A098044.