A379989 -id:A379989 - 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).

30733704, 30733708, 30733714, 30733726, 30733729, 30733733, 30733743, 30733762, 30733764, 30733777, 30733781, 30733796, 30733853, 30733857, 30733860, 30733866, 30733880, 30733887, 30733890, 30734262

Offset: 1

Andrey S. Shchebetov and Sergei D. Shchebetov, Dec 30 2017

nonn

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.
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
The y-coordinate of prime(a(n)) on the Cartesian grid defined in A379643 is -1. - Ya-Ping Lu, Jan 08 2025

Sergei D. Shchebetov, Table of n, a(n) for n = 1..100000
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, 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, Pages 173-197.
Eric Weisstein's World of Mathematics, Prime Quadratic Effect.

Cf. A007350, A007351, A038691, A051024, A051025, A066520, A096628, A096447, A096448, A199547, A275939, A295354, A379643, A379989.

from sympy import nextprime; p, r1, r5 = 1, 0, 0
for n in range(1, 30734263):
    p = nextprime(p); r = p%8
    if r == 1: r1 += 1
    elif r == 5: r5 += 1
if r in {1, 5} and r1 == r5 + 1: print(n, end = ', ')  # Ya-Ping Lu, Jan 08 2025

A380333 Where prime race 8m+3 vs. 8m+7 is tied.

Original entry on oeis.org

2, 7, 31, 37, 41, 47, 53, 79, 103, 127, 199, 223, 239, 241, 263, 269, 283, 293, 311, 313, 317, 367, 373, 383, 389, 397, 401, 409, 431, 433, 443, 449, 457, 461, 467, 499, 523, 541, 1039, 1049, 1063, 1069, 1091, 1093, 1097, 1123, 1129, 1163, 1231, 1237, 1249

Offset: 1

Ya-Ping Lu, Jan 21 2025

nonn

Prime numbers on the y-axis of the Cartesian grid defined in A379643.

Conjecture: There is no prime on the negative y-axis of the Cartesian grid defined in A379643, meaning that prime p does not exist such that pi_{8,3}(p) - pi_{8,7}(p) = 0 and pi_{8,5}(p) - pi_{8,1}(p) < 0, where pi_{m,b}(x) is the number of primes <= x which are congruent to b (mod m).

Cf. A379643, A379731, A379989.

from sympy import nextprime; p, x = 2, 0; R = [p]
while p < 1249:
    p = nextprime(p); d = (5-p%8)//2
    if d in {-1, 1}: x += d
    if x == 0: R.append(p)
print(*R, sep = ', ')

A380877 Primes p where the prime race 12m+1 versus 12m+7 is tied.

Original entry on oeis.org

2, 3, 5, 13, 17, 433, 457, 461

Offset: 1

Ya-Ping Lu, Feb 06 2025

nonn

Primes p such that pi_{12,1}(p) = pi_{12,7}(p), where pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m). For the first 5 billion primes, pi_{12,7}(p) >= pi_{12,1}(p). If exists, a(9) > 122430513841.

Cf. A007351, A068228, A068229, A379989, A380333.

s={};Do[p=Prime[pp];If[Length[Select[Prime[Range[pp]],Mod[#,12]==1&]]==Length[Select[Prime[Range[pp]],Mod[#,12]==7&]],AppendTo[s,p]],{pp,100}];s (* James C. McMahon, Mar 03 2025 *)

from sympy import nextprime; p, d = 2, 0
while p < 500:
    if d == 0: print(p, end = ', ')
    p = nextprime(p); r = p%12
    if r == 7: d += 1
    elif r == 1: d -= 1

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).

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A380333 Where prime race 8m+3 vs. 8m+7 is tied.

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Python

A380877 Primes p where the prime race 12m+1 versus 12m+7 is tied.

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