A215471 Numbers k such that in a rotated-square spiral with positive integers (A215468) among k'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
Keywords
Examples
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.
Links
- David Radcliffe, Table of n, a(n) for n = 1..10391
Programs
-
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, end=' ') -
Python
# See A215468 for the definition of spiral(). from sympy import isprime def neighbors(n): x = A010751(n-1) y = A305258(n-1) return [spiral(x+i, y+j) for i in (-1, 0, 1) for j in (-1, 0, 1) if (i, j) != (0, 0)] def is_A215471(n): return sum(map(isprime, neighbors(n))) >= 5 # David Radcliffe, Aug 05 2025