A379731 List of y coordinates of prime numbers in the Cartesian grid defined in A379643.

0, 0, 1, 1, 1, 2, 1, 1, 1, 2, 2, 3, 2, 2, 2, 3, 3, 4, 4, 4, 3, 3, 3, 2, 1, 2, 2, 2, 3, 2, 2, 2, 1, 1, 2, 2, 3, 3, 3, 4, 4, 5, 5, 4, 5, 5, 5, 5, 5, 6, 5, 5, 4, 4, 3, 3, 4, 4, 5, 4, 4, 5, 5, 5, 4, 5, 5, 4, 4, 5, 4, 4, 4, 5, 5, 5, 6, 7, 6, 5, 5, 6, 6, 5, 5, 5, 4

Offset: 1

Author

Ya-Ping Lu, Dec 31 2024

Keywords

• sign

Comments

The corresponding x coordinates are given in A379643. Positions of the first 1 miilion primes are illustrated in Links in A379643.

Crossrefs

Cf. A297448, A345293, A379643.

Programs

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 = ', ')

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.