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A346948
Isolated single primes enclosed by six composites on hexagonal spiral board of odd numbers.
0
211, 257, 277, 331, 509, 563, 587, 647, 653, 673, 683, 709, 751, 757, 839, 853, 919, 983, 997, 1087, 1117, 1123, 1163, 1283, 1433, 1447, 1493, 1531, 1579, 1637, 1733, 1777, 1889, 1913, 1973, 1993, 2179, 2207, 2251, 2273, 2287, 2333, 2357, 2399, 2447, 2467
OFFSET
1,1
COMMENTS
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.
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:
Region Appearance Arm Appearance
---------------------------------- ---------- ----------------- ----------
6*k^2-18*k+15 <= m <= 6*k^2-16*k+9 2681490 m = 6*k^2-16*k+11 692
6*k^2-16*k+13 <= m <= 6*k^2-14*k+7 3192576 m = 6*k^2-14*k+ 9 551
6*k^2-14*k+11 <= m <= 6*k^2-12*k+5 2681571 m = 6*k^2-12*k+ 7 671
6*k^2-12*k+ 9 <= m <= 6*k^2-10*k+3 2681254 m = 6*k^2-10*k+ 5 545
6*k^2-10*k+ 7 <= m <= 6*k^2- 8*k+1 3191045 m = 6*k^2- 8*k+ 3 721
6*k^2- 8*k+ 5 <= m <= 6*k^2- 6*k-1 2680620 m = 6*k^2- 6*k+ 1 1040
EXAMPLE
3 is not a term because four of the six neighbors (1, 5, 13, 15, 17 and 19) are primes.
211 is a term because 211 is a prime and all six neighbors (145, 147, 209, 213, 287 and 289) are composites.
PROG
(Python)
from sympy import isprime; from math import sqrt, ceil
def neib(m):
if m == 1: return [3, 5, 7, 9, 11, 13]
if m == 3: return [17, 19, 5, 1, 13, 15]
L = [m for i in range(6)]; n = int(ceil((3+sqrt(6*m + 3))/6)); x=6*n*n; y=12*n
a0 = x-18*n+15; a1 =x-16*n+11; a2 =x-14*n+9
a3 = x-y+7; a4 =x-10*n+5; a5 =x-8*n+3; a6 =x-6*n+1
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
L[0] += y-10 if p<=4 else -2 if p<=6 else -y+16 if p<=9 else 2
L[1] += 2 if p<=1 else y-8 if p<=6 else -2 if p<=8 else -y+14
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
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
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
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
return L
for i in range(1, 1500):
m = 2*i - 1
if isprime(m) == 1:
L1 = [neib(m)[j] for j in range(6)]
if sum(isprime(k) for k in L1) == 0: print(m)
KEYWORD
nonn
AUTHOR
Ya-Ping Lu, Aug 08 2021
STATUS
approved