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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 sequence is the complement to A330979; here only composite numbers can be stepped to, while in A330979 only prime numbers can be stepped to. Due to the existence of many more composite numbers than primes the walk here forms a much tigher spiral and generally stays as close as possible to the origin. However the primes occasionally block this preferred path and causes the walk to detour away from the origin, which leaves gaps in the visited squares with composite numbers. Some of these gaps are eventually visited by later steps in the walk.

The first term at which a step to a non-adjacent square is required is a(154) = 74, which steps to a(155) = 158, a distance of sqrt(8) units away. The square with number 74 is surrounded by three primes 43,73,113 and five composites 44,72,75,112,114, all of which have been previously visited.

In the first 1 million terms the longest required step is from a(149464) = 64666, which has coordinates (-127,-22) relative to the starting 1-square, to a(149465) = 67774 with coordinates (-130,-43), a step of length sqrt(450), approximately 21.2 units. See A330782 for the progression of step length records. If the maximum step distance between adjacent composite terms has a finite value or is unbounded as n increases is unknown. The largest difference between adjacent composite terms is for a(650382) = 863400 to a(650383) = 939342, a difference of 75942.

In the first 1 million terms the smallest unvisited composite is 12, which is at coordinates (2,1) relative to the starting square. This square is surrounded by four primes so the walk is never required to step to it during the initial walk steps. See the image in the links. Given the composites become more frequent relative to the primes as n increases it would require a very large detour from the spiral pattern for this square to be visited, so it is likely, although unknown, this square will never be visited. However the link image for 1 million steps shows the path can make detours toward the central square when it is trapped by surrounding paths, so the possibility remains the inner unvisited squares could eventually be visited, although the number of walk steps required before such a detour occurs could be extremely large.

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.

This sequence uses the same rules as A330979 except that, instead of stepping to the closest prime, the path steps to the closest visible square containing a prime i.e., squares containing a prime which have no other square on a line directly between the current position and the square. See A331400 for further details of the visibility of a square on the Ulam spiral.

The restriction of only visiting visible squares containing a prime substantially reduces the possible squares that the walk can step to. Consider the concentric square rings of squares surrounding any square in the Ulam spiral that contains an odd number, as all primes, other than, 2 will be. There are four squares on the adjacent ring of eight squares that are candidates for a visible prime. However on the second square ring of sixteen squares none are candidates as the only visible squares contain even numbers. This should be compared to A330979 where eight of these squares are candidates for the next step. On the third square ring of twenty-four squares only eight squares are candidates, while on the fourth square ring once again there are no candidates as only even numbers are visible. This reduction in nearby candidate squares is reflected by the average step distance for a walk of 10000 steps; in this sequence the average distance is 4.60 units while in A330979 it is 2.98 units.

The first time this sequence differs from A330979 is on the ninth step. A330979(9) = 61 while a(9) = 89. The square with prime 61 is two squares directly to left left of the square a(8) = 59 and is thus blocked from view by the square containing 60, which is one square to the left. The square with prime 89 is at relative coordinates (3,-1) to 59, being the closest visible unvisited prime, and is on the third square ring around 59.

In the first 10 million terms the longest required step is from a(4515899) = 29616101, which has coordinates (-2721,1985) relative to the starting 1-square, to a(4515900) = 28005727 with coordinates (-2646,2184), a step of length sqrt(45226), approximately 212.7 units. 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(9477992) = 132533039 to a(9477993) = 125850199, a difference of 6682840.

In the first 10 million terms the smallest unvisited prime is 571, which has coordinates (-6,12) relative to the starting 1-square. It is unknown if this and similar unvisited prime squares near the origin are eventually visited for very large values of n or are never visited.

The keyword "look" refers to the images in the links. - N. J. A. Sloane, Jun 14 2020

The sequence is infinite as a number containing all ten decimal digits can never be stepped to thus there will always be a square containing a number which has digits not in the number of the current square.

The pattern of visited squares forms nine closely spaced concentric square rings, while these groups of nine have a larger gap of unvisited squares between them. See the linked images.

In the first one million steps the largest single step distance is ~480 units, from a(572017) = 627194 to a(572018) = 3055000. This is a step that jumps between the inner to most outer group of nine concentric rings. The largest single step difference between numbers is from a(721912) = 6951823 to a(721913) = 4404077, a change of 2547746. The smallest unvisited number in the first one million steps is 12, although the image shows the path revisits squares close to the origin after a large number of steps, so it is possible this and other small numbers will eventually be visited.

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.

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.