A335364 -id:A335364 - cp's OEIS frontend

A347358 The prime numbers visited on a square spiral when starting at 1 and then stepping to the smallest unvisited prime number that is visible from the current number.

1, 2, 3, 11, 5, 13, 29, 17, 7, 19, 31, 23, 37, 53, 41, 61, 43, 59, 47, 71, 83, 67, 89, 73, 101, 79, 107, 127, 97, 131, 103, 137, 109, 139, 113, 149, 173, 151, 179, 157, 181, 163, 191, 167, 193, 227, 197, 229, 293, 233, 211, 239, 199, 251, 223, 257, 307, 241, 311, 263, 313, 269, 317, 271, 331, 277

Offset: 1

Scott R. Shannon, Aug 28 2021

nonn

A number is visible from the current number if, given it has coordinates (x,y) relative to the current number, the greatest common divisor of |x| and |y| is 1. See A331400 for the points visible from the starting 1 number.

The primes visited in the sequence appear to oscillate between two different regimes. In one the vast majority of the next smallest visible primes are on the corners of the neighboring inner or outer square ring of numbers, thus the steps are nearly vertical or horizontal relative to the current square. In the other the majority of next smallest visible primes are on square rings much closer or further away from the origin than the current ring, or entirely on the other side of the spiral relative to the starting number. In this regime the path makes very random steps in many different diagonal directions, covering the entire spiral. See the three linked images.

			The square spiral is numbered as follows:
.
  17--16--15--14--13   .
   |               |   .
  18   5---4---3  12   29
   |   |       |   |   |
  19   6   1---2  11   28
   |   |           |   |
  20   7---8---9--10   27
   |                   |
  21--22--23--24--25--26
.
a(1) = 1. The central starting number.
a(2) = 2, a(3) = 3 as 2 is the smallest visible unvisited prime from 1, and 3 is the smallest visible unvisited prime from 2.
a(4) = 11 as 11 is the smallest visible unvisited prime from 3. Note that from 3 the smaller unvisited primes 5 and 7 are hidden from 3 by the numbers 4 and 1.
a(7) = 29 as 29 is the smallest visible unvisited prime from 13. Note that from 13 the smaller unvisited primes 7, 17, 19, 23 are hidden from 13 by numbers 3, 14, 4, 2 respectively.

Scott R. Shannon, Image of the path for the first 7000 terms. The colors are graduated across red, orange, yellow to show the relative step order. Note the yellow lines, terms in the 5000-7000 range, step in all directions across the entire spiral.
Scott R. Shannon, Image of the path for the first 14000 terms. The colors are now graduated across red, orange, yellow, green, blue. Note how the steps for the later colors, terms in the 1000-14000 range, are almost all horizontal or vertical and none step diagonally into the inner spiral.
Scott R. Shannon, Image of the path for the first 21000 terms. The colors are now graduated across red, orange, yellow, green, blue, indigo, violet. Note how the later colors, terms in the 15000-21000 range, again behave like the earlier 5000-7000 term range and step in random directions across the spiral.
Eric Weisstein's World of Mathematics, Visible Point.
Wikipedia, Ulam Spiral.

Cf. A347522 (step to smallest hidden), A000040, A063826, A214664, A214665, A331400, A335364, A332767, A330979.

A341541 a(n) is the number of steps to reach square 1 for a walk starting from square n along the shortest path on the square spiral board without stepping on any prime number. a(n) = -1 if such a path does not exist.

Original entry on oeis.org

0, 1, 2, 1, 2, 1, 2, 1, 2, 3, 4, -1, 4, 3, 2, 3, 4, 19, 2, 17, 16, 15, 2, 3, 4, 5, 4, 5, 6, 11, 6, 5, 4, 3, 4, 5, 6, 19, 18, 19, 18, 17, 16, 15, 14, 13, 4, 5, 6, 7, 6, 5, 6, 9, 10, 11, 10, 9, 6, 5, 4, 5, 6, 7, 8, 9, 10, 17, 18, 19, 18, -1, 16, 15, 14, 13, 12

Offset: 1

Ya-Ping Lu, Feb 14 2021

sign

Conjecture: There is no "island of two or more nonprimes" enclosed by primes on the square spiral board. If the conjecture is true, then numbers n such that a(n) = -1 are the terms in A341542.

			The shortest paths for a(n) <= 20 are illustrated in the figure attached in Links section. If more than one path are available, the path through the smallest number is chosen as the shortest path.

Ya-Ping Lu, Illustration of the shortest paths from n to 1 on the square spiral board without stepping on any prime number

Cf. A341542, A341672, A330979, A332767, A335364, A335661, A336494, A336576.

from sympy import prime, isprime
from math import sqrt, ceil
def neib(m):
    if m == 1: L = [4, 6, 8, 2]
    else:
        n = int(ceil((sqrt(m) + 1.0)/2.0))
        z1 = 4*n*n - 12*n + 10; z2 = 4*n*n - 10*n + 7; z3 = 4*n*n - 8*n + 5
        z4 = 4*n*n - 6*n + 3; z5 = 4*n*n - 4*n + 1
        if m == z1:             L = [m + 1, m - 1, m + 8*n - 9, m + 8*n - 7]
        elif m > z1 and m < z2: L = [m + 1, m - 8*n + 15, m - 1, m + 8*n - 7]
        elif m == z2:           L = [m + 8*n - 5, m + 1, m - 1, m + 8*n - 7]
        elif m > z2 and m < z3: L = [m + 8*n - 5, m + 1, m - 8*n + 13, m - 1]
        elif m == z3:           L = [m + 8*n - 5, m + 8*n - 3, m + 1, m - 1]
        elif m > z3 and m < z4: L = [m - 1, m + 8*n - 3, m + 1, m - 8*n + 11]
        elif m == z4:           L = [m - 1, m + 8*n - 3, m + 8*n - 1, m + 1]
        elif m > z4 and m < z5: L = [m - 8*n + 9, m - 1, m + 8*n - 1, m + 1]
        elif m == z5:           L = [m - 8*n + 9, m - 1, m + 8*n - 1, m + 1]
    return L
step_max = 20; L_last = [1]; L2 = L_last; L3 = [[1]]
for step in range(1, step_max + 1):
    L1 = []
    for j in range(0, len(L_last)):
        m = L_last[j]; k = 0
        while k <= 3 and isprime(m) == 0:
            m_k = neib(m)[k]
            if m_k not in L1 and m_k not in L2: L1.append(m_k)
            k += 1
    L2 += L1; L3.append(L1); L_last = L1
i = 1
while i:
    if isprime(neib(i)[0])*isprime(neib(i)[1])*isprime(neib(i)[2])*isprime(neib(i)[3]) == 1: print(-1)
    elif i not in L2: break
    for j in range(0, len(L3)):
        if i in L3[j]: print(j); break
    i += 1

A336494 The number of steps for a walk on a square spiral numbered board when starting on square 1 and stepping to an unvisited square containing the lowest prime number, where the square is within a block of size (2n+1) X (2n+1) centered on the current square. If no unvisited prime numbered squares exist within the block the walk ends.

Original entry on oeis.org

7, 37, 65, 308, 654, 7214, 21992, 49850, 222791, 1146922, 1912101, 6372680, 23077800

Offset: 1

Scott R. Shannon, Jul 23 2020

For n = 1 this sequence is similar to A335856 except that only prime numbers can be stepped to; if no adjacent prime number exists then the walk ends. In general for a(n) the walk can step to any unvisited square containing the lowest prime number within a block of size (2n+1) X (2n+1) centered on the current square.

See A336576 for the final square number of the walks.

			The board is numbered with the square spiral:
.
  17--16--15--14--13   .
   |               |   .
  18   5---4---3  12   29
   |   |       |   |   |
  19   6   1---2  11   28
   |   |           |   |
  20   7---8---9--10   27
   |                   |
  21--22--23--24--25--26
.
a(1) = 7. Starting from the square 1 the sequence of adjacent unvisited lowest primes the walk can step to are 2,3,11,29,13,31,59. Once the square 59 is visited there are no other unvisited adjacent squares containing primes, so the walk terminates after 7 steps. See the first linked image.
a(2) = 38. This walk also starts by stepping to 2 and then 3. But the next lowest prime 5 is now two units away so is reachable and is thus the next stepped to square. Further steps are 7,19,17,37...,827,829,719,947. Once the square 947 is visited there are no other unvisited squares containing primes within the surrounding 5x5 block of squares, so the walk terminates after 38 steps. See the second linked image.
Also see the linked images for n=3,4,5,6.

Scott R. Shannon, Image showing the 7 steps of the walk for n = 1. For this and other images a green square shows the starting 1 square, a red square shows the final square, and a thick white line is the path between visited squares. All visited prime numbered squares are shown in yellow, while those unvisited squares containing primes are shown in grey. Blue squares mark out the (2n+1)x(2n+1) block of squares around the final square which contains no further unvisited primes. The square spiral numbering of the board is shown as a thin white line.
Scott R. Shannon, Image showing the 37 steps of the walk for n = 2. Click on this and other images to zoom in.
Scott R. Shannon, Image showing the 65 steps of the walk for n = 3.
Scott R. Shannon, Image showing the 308 steps of the walk for n = 4.
Scott R. Shannon, Image showing the 654 steps of the walk for n = 5.
Scott R. Shannon, Image showing the 7214 steps of the walk for n = 6.
Wikipedia, Ulam Spiral.

Cf. A336576 (final square number), A335856, A000040, A136626, A336092, A330979, A332767, A335661, A335364.

A336576 The final square number for a walk on a square spiral numbered board when starting on square 1 and stepping to an unvisited square containing the lowest prime number, where the square is within a block of size (2n+1) x (2n+1) centered on the current square. If no unvisited prime numbered squares exist within the block the walk ends.

Original entry on oeis.org

59, 947, 313, 3331, 5659, 67547, 253801, 676259, 3162413, 16604417, 29135971, 108235159, 437456497

Offset: 1

Scott R. Shannon, Jul 26 2020

See A336494 for an explanation of the sequence and images of the walks.

			a(1) = 59. Starting from the square 1 the sequence of adjacent unvisited lowest primes the walk can step to are 2,3,11,29,13,31,59. Once the square 59 is visited there are no other unvisited adjacent squares containing primes, so the walk terminates.

Wikipedia, Ulam Spiral.

Cf. A336494 (total number of steps), A335856, A000040, A136626, A336092, A330979, A332767, A335661, A335364.

A347522 The prime numbers visited on a square spiral when starting at 1 and then stepping to the smallest unvisited prime number that is not visible from the current number.

Original entry on oeis.org

1, 11, 13, 7, 3, 5, 29, 23, 17, 19, 2, 47, 31, 37, 41, 43, 83, 89, 97, 53, 59, 61, 67, 71, 73, 79, 103, 101, 107, 109, 113, 131, 127, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 229, 227, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 331, 293, 307, 311

Offset: 1

Scott R. Shannon, Sep 05 2021

nonn

A number is not visible from the current number if, given it has coordinates (x,y) relative to the current number, the greatest common divisor of |x| and |y| is greater than 1.

As n increases the vast majority of primes are on the same square ring of numbers as the current prime. However occasionally, especially for primes inside the right side quadrant, the next prime is on an outer or inner ring which causes the step to make a diagonal line. See the linked images. The largest diagonal step after 50000 terms is one at step 43936 between primes 532981 and 531457 which is seen as the long violet diagonal line from the top-left to the bottom-right in the image for these terms. No other such diagonal line is seen up to 10^6 terms.

			The square spiral is numbered as follows:
.
  17--16--15--14--13   .
   |               |   .
  18   5---4---3  12   29
   |   |       |   |   |
  19   6   1---2  11   28
   |   |           |   |
  20   7---8---9--10   27
   |                   |
  21--22--23--24--25--26
.
a(1) = 1. The central starting number.
a(2) = 11 as the smaller prime numbers 2,3,5,7 are all visible from 1, while 11 is hidden by 2.
a(3) = 13 as the smaller prime numbers 2,3,5,7 are all visible from 11, while 13 is hidden by 12.
a(4) = 7 as the smaller prime numbers 2,3,5 are visible from 13, while 7 is hidden by 1 and 3.
a(7) = 29 as the smaller prime numbers 2,17,19,23 are visible from 5, while 29 is hidden by 3,4 and 12.

Scott R. Shannon, Image showing the path taken when connecting the first 1000 terms. The colors are graduated across the spectrum to show the relative step order. The white dots show the visited prime numbers.
Scott R. Shannon, Image showing the path taken when connecting the first 50000 terms.
Eric Weisstein's World of Mathematics, Visible Point.
Wikipedia, Ulam Spiral.

Cf. A347358 (step to smallest visible), A000040, A063826, A214664, A214665, A331400, A335364, A332767, A330979.

A346429 Squares visited on a square spiral when stepping to the closest unvisited square that contains a number with a different number of divisors to the number in the current square. If two or more such squares are the same distance from the current square then the one with the smallest number is chosen.

Original entry on oeis.org

1, 2, 9, 8, 7, 6, 5, 4, 3, 12, 11, 10, 25, 24, 23, 22, 45, 46, 47, 48, 49, 26, 50, 51, 52, 27, 28, 29, 30, 13, 14, 32, 31, 56, 55, 54, 53, 86, 127, 126, 85, 84, 83, 82, 81, 80, 79, 78, 77, 76, 115, 114, 75, 74, 43, 42, 21, 20, 19, 18, 17, 16, 15, 61, 34, 60, 33, 59, 58, 92, 57, 90, 89, 88, 87

Offset: 1

Scott R. Shannon, Jul 17 2021

nonn

The first term at which a step to a non-adjacent square is required is a(64) = 61; the previous square 15 having neighbors already visited or with four divisors.
The linked images show that the path of visited squares can approach the origin after many terms. For example 44 is not visited until the 973644th step, although 43 and 45 are visited after 54 and 16 steps respectively. It is possible eventually all squares are visited although this is unknown.
In the first 10 million terms the longest step distance between terms is on the 8836645th step, between 1548859 and 1578754, a distance of ~90.2 units.

			The square spiral is numbered as follows:
.
  17--16--15--14--13   .
   |               |   .
  18   5---4---3  12   29
   |   |       |   |   |
  19   6   1---2  11   28
   |   |           |   |
  20   7---8---9--10   27
   |                   |
  21--22--23--24--25--26
.
a(3) = 9 as a(2) = 2 which has two divisors, and the closest three unvisited squares around 2 are 3, 11 and 9, and of those only 9 has a divisor count not equal to two.
a(10) = 12 as a(9) = 3 which has two divisors, and the closest two unvisited squares around 3 are 12 and 14. Both have more than two divisors but 12 is the smaller so it the square stepped to.

Scott R. Shannon, Image of the first 5000 steps. The colors are graduated across the spectrum to show the relative step order. Note how after about 4670 step the path approaches the origin again. The central 1 square is marked with a white dot while the smallest unvisited square, 44, is marked with a yellow dot. The longest step distance ~10.6 units after 4681 steps is shown as a white line. Click the image to zoom in.
Scott R. Shannon, Image of the first 1000000 steps. Note the numerous paths that approach the origin. The smallest unvisited square, 159, is marked with a yellow dot. The longest step distance ~43.8 units after 973504 steps is shown as a while line. Click the image to zoom in.

Cf. A335661, A330979, A332767, A335364, A344367.

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A347358 The prime numbers visited on a square spiral when starting at 1 and then stepping to the smallest unvisited prime number that is visible from the current number.

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A341541 a(n) is the number of steps to reach square 1 for a walk starting from square n along the shortest path on the square spiral board without stepping on any prime number. a(n) = -1 if such a path does not exist.

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A347522 The prime numbers visited on a square spiral when starting at 1 and then stepping to the smallest unvisited prime number that is not visible from the current number.

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