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

1,1

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

Links

Table of n, a(n) for n=1..46.

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)

Crossrefs

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

Sequence in context: A246136 A087833 A240918 * A317394 A096706 A001583

Adjacent sequences: A346945 A346946 A346947 * A346949 A346950 A346951

Keyword

nonn

Author

Ya-Ping Lu, Aug 08 2021

Status

approved