OFFSET
1,6
COMMENTS
The corresponding x coordinates are given in A379643. Positions of the first 1 million primes are illustrated in Links in A379643.
From Jianing Song, Jan 08 2026: (Start)
Note that a(n) = -1 for the first time for prime(n) = 588067889 (cf. A297448). Actually, assuming the generalized Riemann hypothesis (GRH) and the Grand Simplicity Hypothesis (GSH; called LI in page 29 of Greg Martin's 2010 link), the set {x>1: pi_{8,5}(x) > pi_{8,1}(x)} has logarithmic density of about 0.9974 (see page 119 of Greg Martin's 2018 link below).
On the other hand, assuming GRH, the sets {x>1: pi_{8,7}(x) > pi_{8,1}(x)} and {x>1: pi_{8,3}(x) > pi_{8,1}(x)} are computed to have logarithmic densities of respectively about 0.9989 and 0.9995 (see pages 120-121 of the 2018 link), and so pi_{8,7}(x) < pi_{8,1}(x) and pi_{8,3}(x) < pi_{8,1}(x) are even more unlikely to happen. (End)
LINKS
Greg Martin and Daniel Fiorilli, Prime number races: An asymptotic formula for the densities, Analytic and Combinatorial Number Theory, Institute of Mathematical Sciences, Chennai, India. August 31, 2010.
Greg Martin, Prime number races, Pure Math Seminar, UNSW Canberra. November 29, 2018.
FORMULA
a(n) = pi_{8,5}(p_n) - pi_{8,1}(p_n), where pi_{m,b}(x) is the number of primes <= x which are congruent to b (mod m) and p_n the n-th prime.
PROG
(Python)
from sympy import nextprime; R = [0, 0]; y, p = 0, 3
for _ in range(85):
p = nextprime(p); d = (p%8 - 3)//2
if d in {-1, 1}: y += d
R.append(y)
print(*R, sep = ', ')
CROSSREFS
KEYWORD
sign
AUTHOR
Ya-Ping Lu, Dec 31 2024
STATUS
approved