OFFSET
1,1
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.
PROG
(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 = ', ')
CROSSREFS
KEYWORD
nonn
AUTHOR
Ya-Ping Lu, Jan 07 2025
STATUS
approved