A215468 -id:A215468 - cp's OEIS frontend

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

Alex Ratushnyak, Aug 11 2012

nonn

			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.

David Radcliffe, Table of n, a(n) for n = 1..10391

Cf. A215468, A215470.

Cf. A039625.

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=' ')

# 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

A217015 Permutation of natural numbers arising from applying the walk of a square spiral (e.g. A214526) to the data of rotated-square spiral (defined in A215468).

Original entry on oeis.org

1, 5, 6, 2, 8, 3, 10, 4, 12, 24, 13, 14, 27, 15, 7, 17, 31, 18, 9, 20, 35, 21, 11, 23, 39, 59, 40, 25, 26, 43, 64, 44, 28, 16, 30, 48, 70, 49, 32, 19, 34, 53, 76, 54, 36, 22, 38, 58, 82, 110, 83, 60, 41, 42, 63, 88, 117, 89, 65, 45, 29, 47, 69, 95, 125, 96, 71, 50

Offset: 1

Alex Ratushnyak, Sep 23 2012

Cf. A090861, A214526, A215468, A217010.

SIZE = 33       # must be 4k+1
grid = [0] * (SIZE*SIZE)
posX = posY = SIZE//2
grid[posY*SIZE+posX]=1
posX += 1
grid[posY*SIZE+posX]=2
n = 3
def walk(stepX, stepY, chkX, chkY):
  global posX, posY, n
  while 1:
    posX+=stepX
    posY+=stepY
    grid[posY*SIZE+posX]=n
    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
import sys
grid2 = [0] * (SIZE*SIZE)
posX = posY = SIZE//2
grid2[posY*SIZE+posX]=1
def walk2(stepX, stepY, chkX, chkY):
  global posX, posY
  while 1:
    a = grid[posY*SIZE+posX]
    if a==0:
        sys.exit(1)
    print(a, end=', ')
    posX+=stepX
    posY+=stepY
    grid2[posY*SIZE+posX]=1
    if grid2[(posY+chkY)*SIZE+posX+chkX]==0:
        return
while 1:
    walk2(0, -1, 1, 0)    # up
    walk2(1, 0, 0, 1)     # right
    walk2(0, 1, -1, 0)    # down
    walk2(-1, 0, 0, -1)   # left

A217296 Permutation of natural numbers arising from applying the walk of rotated-square spiral (defined in A215468) to the data of square spiral (e.g. A214526).

Original entry on oeis.org

1, 4, 6, 8, 2, 3, 15, 5, 19, 7, 23, 9, 11, 12, 14, 34, 16, 18, 40, 20, 22, 46, 24, 10, 28, 29, 13, 33, 61, 35, 17, 39, 69, 41, 21, 45, 77, 47, 25, 27, 53, 54, 30, 32, 60, 96, 62, 36, 38, 68, 106, 70, 42, 44, 76, 116, 78, 48, 26, 52, 86, 87, 55, 31, 59, 95, 139

Offset: 1

Alex Ratushnyak, Sep 30 2012

nonn

Index entries for sequences that are permutations of the natural numbers

Cf. A215468, A214526, A217015.

SIZE = 29    # must be 4k+1
grid = [0] * (SIZE*SIZE)
posX = posY = SIZE//2
grid[posY*SIZE+posX]=1
n = 2
def walk(stepX, stepY, chkX, chkY):
  global posX, posY, n
  while 1:
    posX+=stepX
    posY+=stepY
    grid[posY*SIZE+posX]=n
    n+=1
    if grid[(posY+chkY)*SIZE+posX+chkX]==0:
        return
while posX:
    walk(0, -1, 1, 0)    # up
    walk(1, 0, 0, 1)     # right
    walk(0, 1, -1, 0)    # down
    walk(-1, 0, 0, -1)   # left
grid2 = [0] * (SIZE*SIZE)
posY = SIZE//2
posX = posY+1
grid2[posY*SIZE+posX-1] = grid2[posY*SIZE+posX] = 1
print(1, end=',')
def walk2(stepX, stepY, chkX, chkY):
  global posX, posY
  while 1:
    a = grid[posY*SIZE+posX]
    if a==0:
        raise ValueError
    print(a, end=',')
    posX+=stepX
    posY+=stepY
    grid2[posY*SIZE+posX]=1
    if grid2[(posY+chkY)*SIZE+posX+chkX]==0:
        return
while posX!=SIZE-1:
    walk2(-1,  1, -1, -1)    # down-left
    walk2(-1, -1,  1, -1)    # up-left
    walk2( 1, -1,  1,  0)    # up-right
    walk2( 1,  0,  1,  1)    # right
    walk2( 1,  1, -1,  1)    # down-right

A225590 Primes p such that A217015(p) is a prime number. That is, when applying the walk of a square spiral to the data of rotated-square spiral, on step p a prime number is hit.

Original entry on oeis.org

2, 11, 17, 23, 53, 61, 67, 139, 149, 151, 163, 251, 263, 269, 281, 397, 421, 431, 541, 547, 557, 607, 619, 743, 773, 809, 1021, 1039, 1229, 1279, 1291, 1303, 1361, 1553, 1601, 1619, 1637, 1871, 1901, 1949, 2003, 2239, 2251, 2267, 2281, 2287, 2309, 2311, 2347, 2381, 2393

Offset: 1

Alex Ratushnyak, May 15 2013

nonn

Corresponding primes with prime indices, in sorted order: A225754.

Cf. A217015, A217296, A215468, A214526, A225754.

A225754 Primes p such that A217296(p) is a prime number. That is, when applying the walk of rotated-square spiral to the data of square spiral, on step p a prime number is hit.

Original entry on oeis.org

5, 11, 13, 29, 31, 41, 67, 71, 73, 79, 127, 137, 193, 199, 211, 293, 313, 421, 499, 503, 619, 631, 647, 661, 673, 773, 811, 967, 991, 1013, 1129, 1163, 1553, 1567, 1597, 1601, 1607, 1747, 1777, 1783, 1789, 1801, 1831, 1861, 1997, 2039, 2053, 2087, 2099, 2113, 2287, 2311

Offset: 1

Alex Ratushnyak, May 15 2013

nonn

Corresponding primes with prime indices, in sorted order: A225590. The intersection of a(n) with A225590 begins: 11, 67, 421, 619, 773, 1553, 1601, 2287, 2311, 2381, 2609, 3169, 3181, 3491, 3511, 4157, 4597, 4639, 6263, 7129, 7177, 7193, 8291, 9277.

Cf. A217015, A217296, A215468, A214526, A225590.

cp's OEIS Frontend

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.

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A217015 Permutation of natural numbers arising from applying the walk of a square spiral (e.g. A214526) to the data of rotated-square spiral (defined in A215468).

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A217296 Permutation of natural numbers arising from applying the walk of rotated-square spiral (defined in A215468) to the data of square spiral (e.g. A214526).

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A225590 Primes p such that A217015(p) is a prime number. That is, when applying the walk of a square spiral to the data of rotated-square spiral, on step p a prime number is hit.

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A225754 Primes p such that A217296(p) is a prime number. That is, when applying the walk of rotated-square spiral to the data of square spiral, on step p a prime number is hit.

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