A341542 Numbers on the square spiral board that are enclosed by four primes.

12, 72, 1152, 1452, 1950, 3672, 5520, 6660, 8232, 10302, 10890, 13218, 15288, 15360, 16062, 18042, 20898, 21018, 23628, 25998, 27918, 32190, 37812, 42018, 42462, 48858, 55818, 57192, 80832, 80910, 83340, 91368, 97848, 98640, 104472, 111492, 117498, 119550

Offset: 1

Author

Ya-Ping Lu, Feb 13 2021

Keywords

• nonn

Comments

This sequence is similar to A172294, in which the starting number of the square spiral is 0 instead of 1. For a(n) < 10000000, 4 out of the 782 terms in this sequence, 72, 10302, 415380 and 1624350 are absent in A172294, while 6 out of the 784 terms in A172294, 42, 23562, 83232, 205662, 5805690 and 7019850 are absent in this sequence.

Conjecture: This sequence is infinite. If the conjecture holds, then the twin prime conjecture is true.

The 4 neighbors of n in the spiral are A068225, A068226, A334751, and A334752. - Kevin Ryde, Feb 13 2021

Crossrefs

Cf. A341541, A341672, A172294, A068225, A068226, A334751, and A334752.

Programs

Python

from sympy import isprime
from math import sqrt, ceil
m, m_max = 2, 1000000
while m <= m_max:
    L = [0, 0, 0, 0]
    n = int(ceil((sqrt(m) + 1.0)/2.0))
    z1 = 4*n*n - 12*n + 10
    z2 = 4*n*n - 10*n + 7
    z3 = 4*n*n - 8*n + 5
    z4 = 4*n*n - 6*n + 3
    z5 = 4*n*n - 4*n + 1
    if m > z1 and m < z2: L = [m + 1, m - 8*n + 15, m - 1, m + 8*n - 7]
    elif m > z2 and m < z3: L = [m + 8*n - 5, m + 1, m - 8*n + 13, m - 1]
    elif m > z3 and m < z4: L = [m - 1, m + 8*n - 3, m + 1, m - 8*n + 11]
    elif m > z4 and m < z5: L = [m - 8*n + 9, m - 1, m + 8*n - 1, m + 1]
    if isprime(L[0]) == 1 and isprime(L[1]) == 1 and isprime(L[2]) == 1 and isprime(L[3]) == 1: print(m)
    m += 2