cp's OEIS Frontend

The first term at which a step to a non-adjacent square is required is a(9) = 61; the previous square 59 has adjacent squares 31,32,33,58,60,93,94,95 of which only 31 is prime, but 31 has already been visited at a(7).

In the first 10 million terms the longest required step is from a(8165267) = 22147771, which has coordinates (-2353,1019) relative to the starting 1-square, to a(8165268) = 8236981 with coordinates (-1435,1355), a step of length sqrt(955620), approximately 977.6 units. See A331027 for the progression of step length records. If the maximum step distance between adjacent prime terms has a finite value or is unbounded as n increases is unknown. The largest difference between adjacent prime terms is for a(8176270) = 32960287 to a(8176271) = 18983957, a difference of 13976330.

In the first 10 million terms the smallest unvisited prime is 2701871, which has coordinates (44,822) relative to the starting 1-square. The smallest unvisited term is found to slowly increase as the number of steps increases, indicating that eventually all primes will be visited, although this is unknown. It may require an extremely large number of steps before all primes below a certain value are visited due to the decreasing likelihood of the walk taking the long steps required to visit those primes near the origin which were unvisited in earlier steps.

The eight possible orientations of the Ulam spiral can be derived by combining either A214664 or A214666 with either A214665 or A214667 as ordered pairs of coordinates.

Any even number on the square spiral has 4 diagonally adjacent squares which contain an even number and thus, unless all four such squares have been previously visited, a step to one of those adjacent squares, the one containing the smallest number, will always be possible. Any visited square containing a prime number will need to step to, and be stepped to from, a square containing a multiple of that prime number.

In the first 10 million terms the longest required step is from a(97528) = 5981, a prime number which has coordinates (39,13) relative to the starting 1-square, to a(97529) = 167468 (27*5981), with coordinates (205,-18), a step of length sqrt(28517), approximately 168.9 units. This is an extremely large step length relative to the total number of steps taken up to that point - see the attached link image. It is not surpassed by any subsequent step up to 10 million steps. If the maximum step distance between adjacent terms has a finite value or is unbounded as n increases is unknown. The largest difference between terms is for a(9404208) = 8964653 to a(9404209) = 10485343, a difference of 1520690.

In the first 10 million terms the smallest unvisited square is 37, which has coordinates (-3,3) relative to the starting 1-square. It is unknown if this square, and similar unvisited squares near the origin, is eventually visited for very large values of n or is never visited. The longest run of diagonal steps in the same direction to adjacent smaller even numbers is 52, from a(3979714) = 5051162 to a(3979766) = 4594498.

The sequence A330979 gives the visited primes for a walk on the Ulam Spiral which starts at 1 and then steps to the square containing the closest unvisited prime number. This sequences lists the records for the square of the step distance between primes for that walk. For a walk of 10 million steps the largest square distance is 955620, approximately 977.6 units, which occurs between A330979(8165267) = 22147771, which has coordinates (-2353,1019) relative to the starting 1-square, to A330979(8165268) = 8236981 with coordinates (-1435,1355). See A330979 for an image of the walk. It is unknown if this is a finite or infinite sequence.

This sequences gives the numbers of the squares visited by a chess rook moving on a square-spiral numbered board where the rook starts on the 1 numbered square and at each step moves to the closest unvisited square containing a prime number. The movement is restricted to the four directions a rook can move on a standard chess board, and the rook cannot move over a previously visited square. If two or more unvisited prime numbered squares exist which are the same distance from the current square then the one with the smallest prime number is chosen. Note that if the rook simply moves to the closest unvisited square the sequence will be infinite as the rook will just follow the square spiral path.

The sequence is finite. After 350 steps the square with number 2179 is visited, after which all four squares the rook can move to have been visited.

The first term where this sequence differs from A336447, where the rook steps to the smallest unvisited prime, is a(7) = 43. See the examples below.

The largest visited square is a(151) = 30539. Both the largest step distance between visited squares, 24 units, and the largest prime gap between visited squares, 6744, occur between a(229) = 2143 and a(230) = 8887. The smallest unvisited prime is 11.

This sequences gives the numbers of the squares visited by a chess rook moving on a square-spiral numbered board where the rook starts on the 1 numbered square and at each step moves to an unvisited square containing the smallest prime number. The movement is restricted to the four directions a rook can move on a standard chess board, and the rook cannot move over a previously visited square. Note that if the rook simply moves to an unvisited square containing the smallest number the sequence will be infinite as the rook will just follow the square spiral path.

The sequence is finite. After 134 steps the square with number 863 is visited, after which all four squares the rook can move to have been visited.

The first term where this sequence differs from A336413, where the rook steps to the closest unvisited prime, is a(7) = 37. See the examples below.

The largest visited square is a(102) = 3739. The largest step distance between visited squares is 24 units, between a(128) = 2179 to a(129) = 2203. The largest prime gap between visited squares is 2646, from a(101) = 1093 to a(102) = 3739. The smallest unvisited prime is 11.

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 eight possible orientations of the Ulam spiral can be derived from combining either A214664 or A214666 with either A214665 or A214667 as ordered pairs of coordinates.

This spiral is rotated 180 degrees from the spiral on the March 1964 cover of Scientific American.

The eight possible orientations of the Ulam spiral can be derived from combining either A214664 or A214666 with either A214665 or A214667 as ordered pairs of coordinates.

This spiral is rotated 180 degrees from the spiral on the March 1964 cover of Scientific American.

This sequences gives the numbers of the squares visited by a chess queen moving on a square-spiral numbered board where the queen starts on the 1 numbered square and at each step moves to the closest unvisited square containing a prime number. The movement is restricted to the eight directions a queen can move on a standard chess board, and the queen cannot move over a previously visited square If two or more unvisited prime numbered squares exist which are the same distance from the current square then the one with the smallest prime number is chosen. Note that if the queen simply moves to the closest unvisited square the sequence will be infinite as the queen will just follow the square spiral path.

The sequence is finite. After 519 steps the square with number 1289 is visited, after which all eight squares the queen can move to have been visited.

The first term where this sequence differs from A330979, which steps to the closest unvisited prime without any movement direction restrictions, is a(40) = 227. See the examples below.

The largest visited square is a(292) = 14843. The largest step distance between visited squares is 20 units, between a(338) = 2879 to a(339) = 3779. The largest prime gap between visited squares is 4050, from a(396) = 10667 to a(397) = 14717. The smallest unvisited prime is 41.