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A215470 Prime intersections in a square spiral with positive integers: primes p such that there are four primes among eight nearest neighbors of p. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

LINKS

Table of n, a(n) for n=1..37.

EXAMPLE

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.

PROG

(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,

CROSSREFS

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

Sequence in context: A089163 A343913 A142375 * A344282 A297846 A142304

Adjacent sequences:  A215467 A215468 A215469 * A215471 A215472 A215473

KEYWORD

nonn

AUTHOR

Alex Ratushnyak, Aug 11 2012

STATUS

approved

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Last modified October 3 07:51 EDT 2022. Contains 357231 sequences. (Running on oeis4.)