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

%I A379989 #4 Jan 18 2025 09:29:39
%S A379989 2,3,588065761,588065801,588067729,588067793,588067811,588067849,
%T A379989 588067981,588068773,588068783,588069121,588069149,588069173,
%U A379989 588069179,588069203,588069211,588069259,588069353,588069367,588069401,588069403,588069413,588069431,588069479
%N A379989 Prime numbers on the x-axis of the Cartesian grid defined in A379643.
%C A379989 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).
%C A379989 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).
%C A379989   n              k                         a(n) = p(k)                 x coordinate
%C A379989   -------------  ------------------------  --------------------------  ------------
%C A379989   1              1                         2                           = 0
%C A379989   2              2                         3                           > 0
%C A379989   3    to 1085   30733591   to 31022217    588065761 to 593890729      < 0
%C A379989   1086 to 45606  1531917196 to 1602638733  35615130457 to 37335022091  > 0
%H A379989 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.
%o A379989 (Python)
%o A379989 from sympy import nextprime
%o A379989 p, y = 2, 0; R = [p]
%o A379989 while p < 588069479:
%o A379989     p = nextprime(p); dy = (p%8-3)//2
%o A379989     if dy in {-1, 1}: y += dy
%o A379989     if y == 0: R.append(p)
%o A379989 print(*R, sep = ', ')
%Y A379989 Cf. A297447, A297448, A379643, A379731.
%K A379989 nonn
%O A379989 1,1
%A A379989 _Ya-Ping Lu_, Jan 07 2025