A346948 Isolated single primes enclosed by six composites on hexagonal spiral board of odd numbers.

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

Author

Ya-Ping Lu, Aug 08 2021

Keywords

• nonn

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

Examples

			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.
		

Crossrefs

Cf. A083577, A090687, A176608, A115258, A341542, A344481, A345654.

Programs

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 ma1 and ma2 and ma3 and ma4 and ma5 and m