A379989 Prime numbers on the x-axis of the Cartesian grid defined in A379643.

2, 3, 588065761, 588065801, 588067729, 588067793, 588067811, 588067849, 588067981, 588068773, 588068783, 588069121, 588069149, 588069173, 588069179, 588069203, 588069211, 588069259, 588069353, 588069367, 588069401, 588069403, 588069413, 588069431, 588069479

Offset: 1

Author

Ya-Ping Lu, Jan 07 2025

Keywords

• nonn

Comments

Prime numbers p such that 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).

Terms in this sequence are found in narrow regions covering the sign-changing zones (see Bays and Hudson in Links and Comments in A297448). Terms in the first zone in A297448 are in the region a(3) through a(1085) located on the negative x-axis, while terms in the second zone are in the region a(1086) through a(45606) located on the positive x-axis (see the table below).

n k a(n) = p(k) x coordinate

------------- ------------------------ -------------------------- ------------

1 1 2 = 0

2 2 3 > 0

3 to 1085 30733591 to 31022217 588065761 to 593890729 < 0

1086 to 45606 1531917196 to 1602638733 35615130457 to 37335022091 > 0

Links

• 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.

Crossrefs

Cf. A297447, A297448, A379643, A379731.

Programs

Python

from sympy import nextprime
p, y = 2, 0; R = [p]
while p < 588069479:
    p = nextprime(p); dy = (p%8-3)//2
    if dy in {-1, 1}: y += dy
    if y == 0: R.append(p)
print(*R, sep = ', ')