

A341542


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


6



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
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OFFSET

1,1


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.


LINKS



PROG

(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


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



