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

%I A341542 #8 Feb 18 2021 00:47:45
%S A341542 12,72,1152,1452,1950,3672,5520,6660,8232,10302,10890,13218,15288,
%T A341542 15360,16062,18042,20898,21018,23628,25998,27918,32190,37812,42018,
%U A341542 42462,48858,55818,57192,80832,80910,83340,91368,97848,98640,104472,111492,117498,119550
%N A341542 Numbers on the square spiral board that are enclosed by four primes.
%C A341542 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.
%C A341542 Conjecture: This sequence is infinite. If the conjecture holds, then the twin prime conjecture is true.
%C A341542 The 4 neighbors of n in the spiral are A068225, A068226, A334751, and A334752. - _Kevin Ryde_, Feb 13 2021
%o A341542 (Python)
%o A341542 from sympy import isprime
%o A341542 from math import sqrt, ceil
%o A341542 m, m_max = 2, 1000000
%o A341542 while m <= m_max:
%o A341542     L = [0, 0, 0, 0]
%o A341542     n = int(ceil((sqrt(m) + 1.0)/2.0))
%o A341542     z1 = 4*n*n - 12*n + 10
%o A341542     z2 = 4*n*n - 10*n + 7
%o A341542     z3 = 4*n*n - 8*n + 5
%o A341542     z4 = 4*n*n - 6*n + 3
%o A341542     z5 = 4*n*n - 4*n + 1
%o A341542     if m > z1 and m < z2: L = [m + 1, m - 8*n + 15, m - 1, m + 8*n - 7]
%o A341542     elif m > z2 and m < z3: L = [m + 8*n - 5, m + 1, m - 8*n + 13, m - 1]
%o A341542     elif m > z3 and m < z4: L = [m - 1, m + 8*n - 3, m + 1, m - 8*n + 11]
%o A341542     elif m > z4 and m < z5: L = [m - 8*n + 9, m - 1, m + 8*n - 1, m + 1]
%o A341542     if isprime(L[0]) == 1 and isprime(L[1]) == 1 and isprime(L[2]) == 1 and isprime(L[3]) == 1: print(m)
%o A341542     m += 2
%Y A341542 Cf. A341541, A341672, A172294, A068225, A068226, A334751, and A334752.
%K A341542 nonn
%O A341542 1,1
%A A341542 _Ya-Ping Lu_, Feb 13 2021