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%I #15 Aug 13 2018 09:11:58
%S 71,353,701,1151,1451,3347,4691,13463,21017,27947,34337,42017,52253,
%T 57191,79907,80831,81611,121469,144497,159737,161141,256301,265547,
%U 284231,285707,312161,334511,346559,348617,382601,392069,422867,440303,502013,541061,545873,593207
%N Prime intersections in a square spiral with positive integers: primes p such that there are four primes among eight nearest neighbors of p.
%C Conjecture: the sequence is infinite. - _Alex Ratushnyak_, Sep 19 2012
%e The spiral begins:
%e .
%e 121 82--83--84--85--86--87--88--89--90--91
%e | | |
%e 120 81 50--51--52--53--54--55--56--57 92
%e | | | | |
%e 119 80 49 26--27--28--29--30--31 58 93
%e | | | | | | |
%e 118 79 48 25 10--11--12--13 32 59 94
%e | | | | | | | | |
%e 117 78 47 24 9 2---3 14 33 60 95
%e | | | | | | | | | | |
%e 116 77 46 23 8 1 4 15 34 61 96
%e | | | | | | | | | |
%e 115 76 45 22 7---6---5 16 35 62 97
%e | | | | | | | |
%e 114 75 44 21--20--19--18--17 36 63 98
%e | | | | | |
%e 113 74 43--42--41--40--39--38--37 64 99
%e | | | |
%e 112 73--72--71--70--69--68--67--66--65 100
%e | |
%e 111-110-109-108-107-106-105-104-103-102-101
%e .
%e Among eight nearest neighbors of 71 four are primes: 41, 43, 107, 109.
%o (Python)
%o SIZE = 3335 # must be odd
%o TOP = SIZE*SIZE
%o prime = [1]*TOP
%o prime[1]=0
%o for i in range(4,TOP,2):
%o prime[i]=0
%o for i in range(3,TOP,2):
%o if prime[i]==1:
%o for j in range(i*3,TOP,i*2):
%o prime[j]=0
%o grid = [0] * TOP
%o posX = posY = SIZE//2
%o grid[posY*SIZE+posX] = 1
%o n = 2
%o saveX = [0]* (TOP+1)
%o saveY = [0]* (TOP+1)
%o saveX[1]=posX
%o saveY[1]=posY
%o def walk(stepX, stepY, chkX, chkY):
%o global posX, posY, n
%o while 1:
%o posX+=stepX
%o posY+=stepY
%o grid[posY*SIZE+posX]=n
%o saveX[n]=posX
%o saveY[n]=posY
%o n+=1
%o if posX*posY==0 or grid[(posY+chkY)*SIZE+posX+chkX]==0:
%o return
%o while 1:
%o walk(0, -1, 1, 0) # up
%o if posX*posY==0:
%o break
%o walk(1, 0, 0, 1) # right
%o walk(0, 1, -1, 0) # down
%o walk(-1, 0, 0, -1) # left
%o for s in range(1, n):
%o if prime[s]:
%o posX = saveX[s]
%o posY = saveY[s]
%o a,b=(grid[(posY-1)*SIZE+posX-1]) , (grid[(posY-1)*SIZE+posX+1])
%o c,d=(grid[(posY+1)*SIZE+posX-1]) , (grid[(posY+1)*SIZE+posX+1])
%o if a*b==0 or c*d==0:
%o break
%o if prime[a]+prime[b]+prime[c]+prime[d]==4:
%o print s,
%Y Cf. A137928, A137930, A137931, A114254, A214176, A214177, A215471.
%K nonn
%O 1,1
%A _Alex Ratushnyak_, Aug 11 2012
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