A215470 Prime intersections in a square spiral with positive integers: primes p such that there are four primes among eight nearest neighbors of p.

71, 353, 701, 1151, 1451, 3347, 4691, 13463, 21017, 27947, 34337, 42017, 52253, 57191, 79907, 80831, 81611, 121469, 144497, 159737, 161141, 256301, 265547, 284231, 285707, 312161, 334511, 346559, 348617, 382601, 392069, 422867, 440303, 502013, 541061, 545873, 593207

Offset: 1

Author

Alex Ratushnyak, Aug 11 2012

Keywords

• nonn

Comments

Conjecture: the sequence is infinite. - Alex Ratushnyak, Sep 19 2012

Examples

			The spiral begins:
.
  121  82--83--84--85--86--87--88--89--90--91
    |   |                                   |
  120  81  50--51--52--53--54--55--56--57  92
    |   |   |                           |   |
  119  80  49  26--27--28--29--30--31  58  93
    |   |   |   |                   |   |   |
  118  79  48  25  10--11--12--13  32  59  94
    |   |   |   |   |           |   |   |   |
  117  78  47  24   9   2---3  14  33  60  95
    |   |   |   |   |   |   |   |   |   |   |
  116  77  46  23   8   1   4  15  34  61  96
    |   |   |   |   |       |   |   |   |   |
  115  76  45  22   7---6---5  16  35  62  97
    |   |   |   |               |   |   |   |
  114  75  44  21--20--19--18--17  36  63  98
    |   |   |                       |   |   |
  113  74  43--42--41--40--39--38--37  64  99
    |   |                               |   |
  112  73--72--71--70--69--68--67--66--65 100
    |                                       |
  111-110-109-108-107-106-105-104-103-102-101
.
Among eight nearest neighbors of 71 four are primes: 41, 43, 107, 109.
		

Crossrefs

Cf. A137928, A137930, A137931, A114254, A214176, A214177, A215471.

Programs

Python

SIZE = 3335  # must be odd
TOP = SIZE*SIZE
prime = [1]*TOP
prime[1]=0
for i in range(4,TOP,2):
    prime[i]=0
for i in range(3,TOP,2):
    if prime[i]==1:
        for j in range(i*3,TOP,i*2):
            prime[j]=0
grid = [0] * TOP
posX = posY = SIZE//2
grid[posY*SIZE+posX] = 1
n = 2
saveX = [0]* (TOP+1)
saveY = [0]* (TOP+1)
saveX[1]=posX
saveY[1]=posY
def walk(stepX, stepY, chkX, chkY):
  global posX, posY, n
  while 1:
    posX+=stepX
    posY+=stepY
    grid[posY*SIZE+posX]=n
    saveX[n]=posX
    saveY[n]=posY
    n+=1
    if posX*posY==0 or grid[(posY+chkY)*SIZE+posX+chkX]==0:
        return
while 1:
    walk(0, -1, 1, 0)   # up
    if posX*posY==0:
        break
    walk(1, 0, 0, 1)    # right
    walk(0, 1, -1, 0)   # down
    walk(-1, 0, 0, -1)  # left
for s in range(1, n):
  if prime[s]:
    posX = saveX[s]
    posY = saveY[s]
    a,b=(grid[(posY-1)*SIZE+posX-1]) , (grid[(posY-1)*SIZE+posX+1])
    c,d=(grid[(posY+1)*SIZE+posX-1]) , (grid[(posY+1)*SIZE+posX+1])
    if a*b==0 or c*d==0:
        break
    if prime[a]+prime[b]+prime[c]+prime[d]==4:
        print(s, end=', ')