cp's OEIS Frontend

A297447 Values of n for which pi_{8,5}(p_n) - pi_{8,1}(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).

%I A297447 #27 May 27 2025 10:09:14
%S A297447 30733704,30733708,30733714,30733726,30733729,30733733,30733743,
%T A297447 30733762,30733764,30733777,30733781,30733796,30733853,30733857,
%U A297447 30733860,30733866,30733880,30733887,30733890,30734262
%N A297447 Values of n for which pi_{8,5}(p_n) - pi_{8,1}(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 A297447 This is a companion sequence to A297448. The first two sign-changing zones were discovered by Bays and Hudson back in 1979. We discovered four additional zones starting from a(22794) = 186422420112. The full sequence with all 6 zones checked up to 5*10^14 contains 664175 terms (see a-file) with a(664175) = 6097827689926 as its last term.
%C A297447 This sequence was checked up to 10^15 and the 7th sign-changing zone starting from a(664176) = 27830993289634 and ending with a(850232)= 27876113171315 was found. - Andrey S. Shchebetov and _Sergei D. Shchebetov_, Jul 28 2018
%C A297447 The y-coordinate of prime(a(n)) on the Cartesian grid defined in A379643 is -1. - _Ya-Ping Lu_, Jan 08 2025
%H A297447 Sergei D. Shchebetov, <a href="/A297447/b297447.txt">Table of n, a(n) for n = 1..100000</a>
%H A297447 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 A297447 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 A297447 M. Deléglise, P. Dusart, and 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 A297447 A. Granville and G. Martin, <a href="https://web.archive.org/web/20240529054811/https://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 A297447 M. Rubinstein and 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 A297447 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeQuadraticEffect.html">Prime Quadratic Effect.</a>
%o A297447 (Python)
%o A297447 from sympy import nextprime; p, r1, r5 = 1, 0, 0
%o A297447 for n in range(1, 30734263):
%o A297447     p = nextprime(p); r = p%8
%o A297447     if r == 1: r1 += 1
%o A297447     elif r == 5: r5 += 1
%o A297447 if r in {1, 5} and r1 == r5 + 1: print(n, end = ', ')  # _Ya-Ping Lu_, Jan 08 2025
%Y A297447 Cf. A007350, A007351, A038691, A051024, A051025, A066520, A096628, A096447, A096448, A199547, A275939, A295354, A379643, A379989.
%K A297447 nonn
%O A297447 1,1
%A A297447 Andrey S. Shchebetov and _Sergei D. Shchebetov_, Dec 30 2017