A215471 Numbers n such that in a rotated-square spiral with positive integers (A215468) among n's eight nearest neighbors five or more are primes.

8, 30, 138, 658, 2620, 3010, 3168, 3372, 3462, 8628, 11940, 17682, 24918, 27918, 32560, 39228, 39790, 40128, 43608, 48532, 53268, 55372, 56040, 56712, 73362, 85200, 85888, 90646, 96052, 101748, 102652, 104382, 112068, 113932, 115330, 119298, 128518, 129288, 131500

Offset

1,1

Links

Table of n, a(n) for n=1..39.

Example

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

Prog

(Python)
SIZE = 3335  # must be odd
TOP = SIZE*SIZE
t = TOP//2
prime = [1]*t
prime[1]=0
for i in range(4, t, 2):
    prime[i]=0
for i in range(3, t, 2):
    if prime[i]==1:
        for j in range(i*3, t, i*2):
            prime[j]=0
grid = [0] * TOP
posX = posY = SIZE//2
saveX = [0]* (t+1)
saveY = [0]* (t+1)
grid[posY*SIZE+posX] = 1
saveX[1]=posX
saveY[1]=posY
posX += 1
grid[posY*SIZE+posX] = 2
saveX[2]=posX
saveY[2]=posY
n = 3
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 grid[(posY+chkY)*SIZE+posX+chkX]==0:
        return
while posX!=SIZE-1:
    walk(-1,  1, -1, -1)    # down-left
    walk(-1, -1,  1, -1)    # up-left
    walk( 1, -1,  1,  0)    # up-right
    walk( 1,  0,  1,  1)    # right
    walk( 1,  1, -1,  1)    # down-right
for s in range(1, n):
    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])
    e, f=(grid[(posY-1)*SIZE+posX  ]) , (grid[(posY+1)*SIZE+posX  ])
    g, h=(grid[ posY   *SIZE+posX-1]) , (grid[ posY   *SIZE+posX+1])
    if a*b==0 or c*d==0 or e*f==0 or g*h==0:
        break
    z = prime[a]+prime[b]+prime[c]+prime[d]
    if z+prime[e]+prime[f]+prime[g]+prime[h] >= 5:
        print s,

Crossrefs

Cf. A215468, A215470.

Cf. A039625.

Sequence in context: A293965 A161222 A229374 * A221145 A098741 A027146

Adjacent sequences: A215468 A215469 A215470 * A215472 A215473 A215474

Keyword

nonn

Author

Alex Ratushnyak, Aug 11 2012

Status

approved