A345654 Numbers with five neighboring primes on the hexagonal spiral board of odd numbers.

1, 15, 45, 63, 165, 195, 231, 459, 693, 909, 969, 1299, 1785, 2709, 3699, 4131, 4449, 5145, 7041, 8541, 10209, 16065, 20355, 22569, 27489, 28299, 38151, 47745, 49365, 49959, 58479, 77619, 81021, 84651, 87555, 92625, 101115, 104181, 107271, 107349, 108225

Offset: 1

Author

Ya-Ping Lu, Jun 21 2021

Keywords

• nonn

Comments

All terms in this sequence are composites.

Conjecture: This sequence is infinite and, except 1 and 15, all terms appear in the region between 6*k^2-16*k+11 and 6*k^2-14*k+9 or between 6*k^2-10*k+5 and 6*k^2-8*k+3, where k (>= 1) is the layer number on the hexagonal board.

If the conjecture is true, twin prime conjecture follows.

Examples

			1 is a term because five of its six neighbors (3, 5, 7, 9, 11, and 13) are primes;
45 is a term because five of its six neighbors (17, 19, 43, 47, 83, and 85) are primes.
A hexagonal spiral board of odd numbers <= 169 is illustrated in the figure below, where terms in the sequence are shown in square brackets and primes in parentheses.
.
                 (151)<(149)<-147<--145<--143<--141
                   /                               \
                  /                                 \
               153   (97)<--95<---93<---91<--(89) (139)
                /     /                         \     \
               /     /                           \     \
            155    99    55<--(53)<--51<---49    87  (137)
             /     /     /                   \     \     \
            /     /     /                     \     \     \
        (157) (101)   57    25<--(23)<--21   (47)   85   135
          /     /     /     /             \     \     \     \
         /     /     /     /               \     \     \     \
      159  (103)  (59)   27    (7)<--(5)  (19)  [45]  (83)  133
       /     /     /     /     /       \     \     \     \     \
      /     /     /     /     /         \     \     \     \     \
   161   105   (61)  (29)    9    [1]-->(3)  (17)  (43)   81  (131)
      \     \     \     \     \               /     /     /     /
       \     \     \     \     \             /     /     /     /
     (163) (107)  [63]  (31)  (11)->(13)->[15]  (41)  (79)  129
         \     \     \     \                     /     /     /
          \     \     \     \                   /     /     /
        [165] (109)   65    33--->35-->(37)-->39    77  (127)
            \     \     \                           /     /
             \     \     \                         /     /
           (167)  111   (67)-->69-->(71)->(73)-->75   125
               \     \                                 /
                \     \                               /
               169  (113)->115-->117-->119-->121-->123
		

Crossrefs

Cf. A341542.

Programs

Python

from sympy import isprime; from math import sqrt, ceil
def neib(m):
    if m == 1: L = [3, 5, 7, 9, 11, 13]
    elif m == 3: L = [17, 19, 5, 1, 13, 15]
    else:
        L = [m for i in range(6)]; n = int(ceil((3+sqrt(6*m+3))/6))
        a0=6*n*n-18*n+15; a1=6*n*n-16*n+11; a2=6*n*n-14*n+9; a3=6*n*n-12*n+7; a4=6*n*n-10*n+5; a5=6*n*n-8*n+3; a6=6*n*n-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