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%I #9 Oct 03 2021 19:20:20
%S 211,257,277,331,509,563,587,647,653,673,683,709,751,757,839,853,919,
%T 983,997,1087,1117,1123,1163,1283,1433,1447,1493,1531,1579,1637,1733,
%U 1777,1889,1913,1973,1993,2179,2207,2251,2273,2287,2333,2357,2399,2447,2467
%N Isolated single primes enclosed by six composites on hexagonal spiral board of odd numbers.
%C It seems that more isolated primes, m, appear in regions 6*k^2-16*k+13 <= m <= 6*k^2-14*k+7 and 6*k^2-10*k+7 <= m <= 6*k^2-8*k+1 than the other 4 regions, where k (>= 1) is the layer number on the hexagonal board, which is illustrated in A345654.
%C Numbers of prime terms appearing in the 6 regions and 6 arms of a 10000-layer hexagonal board, with the 299970001 odd numbers up to 599940001, are:
%C Region Appearance Arm Appearance
%C ---------------------------------- ---------- ----------------- ----------
%C 6*k^2-18*k+15 <= m <= 6*k^2-16*k+9 2681490 m = 6*k^2-16*k+11 692
%C 6*k^2-16*k+13 <= m <= 6*k^2-14*k+7 3192576 m = 6*k^2-14*k+ 9 551
%C 6*k^2-14*k+11 <= m <= 6*k^2-12*k+5 2681571 m = 6*k^2-12*k+ 7 671
%C 6*k^2-12*k+ 9 <= m <= 6*k^2-10*k+3 2681254 m = 6*k^2-10*k+ 5 545
%C 6*k^2-10*k+ 7 <= m <= 6*k^2- 8*k+1 3191045 m = 6*k^2- 8*k+ 3 721
%C 6*k^2- 8*k+ 5 <= m <= 6*k^2- 6*k-1 2680620 m = 6*k^2- 6*k+ 1 1040
%e 3 is not a term because four of the six neighbors (1, 5, 13, 15, 17 and 19) are primes.
%e 211 is a term because 211 is a prime and all six neighbors (145, 147, 209, 213, 287 and 289) are composites.
%o (Python)
%o from sympy import isprime; from math import sqrt, ceil
%o def neib(m):
%o if m == 1: return [3, 5, 7, 9, 11, 13]
%o if m == 3: return [17, 19, 5, 1, 13, 15]
%o L = [m for i in range(6)]; n = int(ceil((3+sqrt(6*m + 3))/6)); x=6*n*n; y=12*n
%o a0 = x-18*n+15; a1 =x-16*n+11; a2 =x-14*n+9
%o a3 = x-y+7; a4 =x-10*n+5; a5 =x-8*n+3; a6 =x-6*n+1
%o p = 0 if m==a0 else 1 if m>a0 and m<a1 else 2 if m==a1 else 3 if m>a1 and m<a2 else 4 if m==a2 else 5 if m>a2 and m<a3 else 6 if m==a3 else 7 if m>a3 and m<a4 else 8 if m==a4 else 9 if m>a4 and m<a5 else 10 if m==a5 else 11 if m>a5 and m<a6 else 12
%o L[0] += y-10 if p<=4 else -2 if p<=6 else -y+16 if p<=9 else 2
%o L[1] += 2 if p<=1 else y-8 if p<=6 else -2 if p<=8 else -y+14
%o L[2] += -y+24 if p<=1 else 2 if p<=3 else y-6 if p<=8 else -2 if p<=10 else -y+12
%o L[3] += -2 if p==0 else -y+22 if p<=3 else 2 if p<=5 else y-4 if p<=10 else -2
%o L[4] += y-14 if p==0 else -2 if p<=2 else -y+20 if p<=5 else 2 if p<=7 else y-2
%o L[5] += y-12 if p<=2 else -2 if p<=4 else -y+18 if p<=7 else 2 if p<=9 else y
%o return L
%o for i in range(1, 1500):
%o m = 2*i - 1
%o if isprime(m) == 1:
%o L1 = [neib(m)[j] for j in range(6)]
%o if sum(isprime(k) for k in L1) == 0: print(m)
%Y Cf. A083577, A090687, A176608, A115258, A341542, A344481, A345654.
%K nonn
%O 1,1
%A _Ya-Ping Lu_, Aug 08 2021
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