Displaying 1-10 of 23 results found.
The records for distance squared for step lengths between adjacent primes in A330979, the visited primes for a walk stepping to the closest unvisited prime on the Ulam Spiral.
+20
10
1, 2, 4, 8, 10, 32, 74, 90, 136, 290, 360, 388, 394, 674, 802, 872, 1184, 1394, 3316, 4100, 5300, 5634, 10728, 23098, 25128, 26836, 33508, 53954, 61092, 66610, 92858, 187540, 190120, 215104, 217732, 955620
COMMENTS
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.
EXAMPLE
The below table shows the details of the record step lengths of this sequence. The coordinate is relative to the starting 1-square.
--------------------------------------------------------------------------------
a(n) | A330979 step # | Start prime & coord | End prime & coord | --------------------------------------------------------------------------------
1 | 1 | 1 (0,0) | 2 (1,0) |
2 | 3 | 3 (1,1) | 11 (2,0) |
4 | 8 | 59 (2,4) | 61 (0,4) |
8 | 14 | 193 (-3,7) | 101 (-5,5) |
10 | 38 | 167 (4,-6) | 83 (5,-3) |
32 | 59 | 631 (13,7) | 1103 (17,3) |
74 | 169 | 113 (-3,-5) | 53 (4,0) |
90 | 319 | 17239 (66,12) | 22291 (75,15) |
136 | 1152 | 2719 (-26,12) | 4127 (-32,2) |
290 | 1659 | 13187 (19,-57) | 7907 (30,-44) |
360 | 2607 | 45263 (0,-106) | 40283 (-18,-100) |
388 | 7397 | 29723 (-86,-52) | 35509 (-94,-70) |
394 | 7806 | 47653 (-109,-19) | 59663 (-122,-4) |
674 | 7877 | 83101 (-144,-12) | 114419 (-169,-5) |
802 | 24920 | 2637497 (-812,692) | 2515477 (-793,713) |
872 | 27038 | 1285799 (-409,567) | 1170607 (-423,541) |
1184 | 55427 | 720089 (-288,-424) | 653761 (-316,-404) |
1394 | 56478 | 460349 (-339,-325) | 457687 (-304,-338) |
3316 | 56480 | 452293 (-300,-336) | 410203 (-320,-282) |
4100 | 82533 | 156353 (198,130) | 129263 (158,180) |
5300 | 83192 | 394211 (-140,314) | 331697 (-208,288) |
5634 | 165879 | 63589 (-126,42) | 161761 (-201,45) |
10728 | 237806 | 1034387 (509,-411) | 962543 (491,-309) |
23098 | 556765 | 110603 (-120,-166) | 19249 (-3,-69) |
25128 | 770967 | 7070333 (1330,-1278) | 8614337 (1468,-1356) |
26836 | 1074758 | 3213377 (-576,-896) | 3582083 (-420,-946) |
33508 | 1074809 | 4140079 (-129,-1017) | 2995469 (-27,-865) |
53954 | 2257389 | 67480409 (-1709,-4107) | 72669481 (-1882,-4262) |
61092 | 2644510 | 5269679 (790,1148) | 5492621 (544,1172) |
66610 | 2644988 | 1156873 (366,538) | 694591 (417,285) |
92858 | 2669627 | 109789 (166,62) | 122443 (-117,175) |
187540 | 2730402 | 2509621 (-792,228) | 672787 (-410,24) |
190120 | 2730411 | 193771 (-220,50) | 296827 (74,-272) |
215104 | 2730444 | 1505201 (285,-613) | 4506473 (405,-1061) |
217732 | 8160823 | 61908241 (-3934,1118) | 48110423 (-3468,1142) |
955620 | 8165267 | 22147771 (-2353,1019) | 8236981 (-1435,1355) |
The squares visited on the 2D square (Ulam) spiral when starting at square 1 and then stepping to the closest unvisited square which contains a composite number. If two or more squares are the same distance from the current square then the one with the smallest composite number is chosen.
+10
19
1, 4, 15, 14, 33, 32, 30, 55, 54, 87, 86, 85, 52, 27, 10, 9, 8, 6, 18, 39, 38, 36, 35, 16, 34, 60, 95, 94, 93, 58, 57, 56, 88, 129, 128, 177, 176, 175, 126, 125, 84, 51, 26, 25, 24, 46, 45, 22, 21, 20, 40, 69
COMMENTS
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.
LINKS
Scott R. Shannon, Illustration of a section of the walk up to n = 450. This shows how the square with number 12, which has four adjacent primes 1 unit away, is not visited during the initial part of the walk. Various other unvisited composites can also be seen. Scott R. Shannon, Illustration of the walk up to n = 1000000. The color of each step is graduated across the spectrum from red to violet to show the relative visit order of the squares. The starting square is shown as a white dot and the smallest unvisited composite square with number 12 is shown as a yellow dot. Note the walk steps shown in yellow which make a detour toward the central squares after about 150,000 steps. Click on the image to zoom in.
EXAMPLE
a(2) = 4 as the starting square numbered 1 has three adjacent squares 1 unit away with numbers 4,6,8, and 4 is the smallest number of those.
a(4) = 14 as the previous visited square 15 has three unvisited adjacent composite number 14,16,34, and 14 is the smallest number of those.
a(7) = 30 as the previous number 32 is has three primes and one visited composite square one unit away. The next closest unvisited composites, sqrt(2) units away, are 30,58,60, and 30 is the smallest of those.
The squares visited on a square (Ulam) spiral, with a(1) = 1 and a(2) = 2, when stepping to the closest unvisited square containing a number that shares a common divisor > 1 with 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.
+10
12
1, 2, 4, 6, 8, 22, 20, 40, 18, 39, 69, 105, 150, 104, 66, 38, 36, 63, 98, 62, 34, 14, 12, 3, 15, 5, 35, 60, 33, 30, 55, 88, 54, 87, 129, 177, 234, 299, 455, 375, 456, 374, 300, 235, 130, 90, 57, 93, 135, 186, 134, 92, 58, 32, 56, 91, 133, 182, 132, 180, 237
COMMENTS
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.
LINKS
Scott R. Shannon, Image of the steps from 1 to 20001. The green dot shows the starting square 1, the red dot the final square 26453, and the yellow dot the smallest unvisited square 11. The orange line shows the largest step distance, sqrt(976), from a(8538) = 233 to a(8539) = 3029. The blue line shows the longest run of adjacent diagonal steps, each of length sqrt(2), to a lower even number in the same direction, from a(3747) = 7880 and lasts for 19 steps. The pink line shows the largest change in value for a single step, from a(19032) = 15023 to a(19033) = 25159, a difference of 10136. Scott R. Shannon, Image of the steps from 1 to 20001 with color. The color of each step is graduated across the spectrum from red to violet to show the relative visit order of the squares. Note how green colored steps, those around n = 10000, approach the origin, showing that all numbers near the origin may eventually be visited for very large values of n. Scott R. Shannon, Image of the steps from 1 to 100000. The orange line shows the step length of sqrt(28517) units at a(97528), from 5981 to 167468. The blue line shows the new longest run of adjacent diagonal steps to lower even numbers, a series of 24 steps. The yellow dot shows the new lowest unvisited square 13, square 11 being visited at a(26321). Scott R. Shannon, Image of the steps from 1 to 5000000 with color. Note how some violet colored steps, those around n = 4200000, approach the origin. The yellow dot shows the new lowest unvisited square 37, square 13 being visited at a(105263). Also note the visited area forms a roughly square pattern, following the largest outer numbers of the spiral. This becomes more pronounced as n increases.
EXAMPLE
a(3) = 4 as a(2) = 2 is surrounded by eight adjacent squares with numbers 3,4,1,8,9,10,11,12. The unvisited squares 1 unit away, 3,9,11 have no common factor with 2. Of the other squares sqrt(2) units away, 4,8,10,12, all share the common factor 2 with a(2), and the smallest of those is 4.
a(10) = 39 as a(9) = 18 is surrounded by adjacent squares 5,6,19,40,39,38,17,16. The square containing 39 is 1 unit directly left of 18 and shares the common factor 3. The other squares one unit away, 5,17,19, have no common factor with 18.
The squares visited on the Ulam spiral when starting at square 1 and then stepping to the closest visible unvisited square which contains a prime number. If two or more visible squares are the same distance from the current square then the one with the smallest prime number is chosen.
+10
7
1, 2, 3, 11, 29, 13, 31, 59, 89, 131, 179, 127, 83, 53, 5, 17, 37, 67, 103, 149, 101, 61, 97, 139, 191, 251, 193, 137, 313, 389, 311, 241, 307, 379, 461, 383, 467, 557, 463, 761, 653, 757, 647, 751, 863, 983, 643, 547, 457, 239, 181, 233, 173, 229, 293, 227, 223, 167, 521, 433, 353, 281
COMMENTS
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
LINKS
Scott R. Shannon, Image for the steps from n = 1 to 20001 with color. The starting square a(1) = 1 is shown as a white dot and the square a(20001) = 220019 is shown as a red dot. The smallest unvisited prime after 20000 steps, 107, is shown as a yellow dot. The color of each step is graduated across the spectrum from red to violet to show the relative visit order of the squares. Scott R. Shannon, Image for the steps from n = 1 to 5000000 with color. Note that some violet colored steps, corresponding to n values over 4000000, approach the origin, indicating earlier unvisited prime squares near the origin may eventually be visited after a large number of steps.
Squares visited on a spirally numbered board when stepping to the closest unvisited square which contains a number that shares no digit with the number of the current square. If two or more such squares are the same distance away the one with the smaller number is chosen.
+10
6
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 25, 48, 79, 80, 49, 26, 51, 84, 125, 83, 50, 81, 52, 86, 53, 28, 11, 27, 85, 126, 87, 54, 29, 30, 55, 88, 129, 56, 31, 58, 93, 57, 90, 131, 89, 130, 92, 135, 94, 137, 95, 60, 33, 14, 32, 59, 13, 62, 35, 16, 34, 15, 36, 17, 38, 67, 104, 66, 37, 64, 99, 100, 65, 102
COMMENTS
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.
LINKS
Scott R. Shannon, Image of the first 6000 steps. The step colors are graduated across the spectrum from red to violet to show the relative step ordering. The starting square is shown as a white dot.
EXAMPLE
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(2) = 2 as from 1 there are four numbers one unit away, 2,4,6,8, none of which contain the digit 1, so of these the smallest is chosen, which is 2.
a(11) = 25 as from the square 10 the square with 25 is only one unit away and shares no digit with 10.
a(20) = 83 as the four squares one unit away from 125 have been visited or contain digits 1,2 or 5. The square with 83 is diagonally adjacent to 125 and is the first time a square more than one unit away is stepped to.
a(23) = 52, and is the first square stepped to that is not adjacent to the previous square, being three units away from 81. All closer squares have been either visited or contain a 1 or 8 in their number.
Squares visited on a spirally numbered board when stepping to the closest unvisited square that contains a number that shares one or more digits with the number of the current square. If two or more such squares are the same distance away the one with the smaller number is chosen.
+10
6
1, 11, 10, 12, 13, 14, 15, 16, 17, 18, 19, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 3, 23, 22, 21, 20, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 2, 52, 51, 50, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 59, 58, 57, 56, 55, 54, 53, 125, 124, 123, 122, 121, 120
LINKS
Scott R. Shannon, Image of the first 1000 steps.. The colors are graduated across the spectrum to show the relative step order. The lowest unvisited square, 4, is marked with a yellow dot.
EXAMPLE
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(2) = 11. There are three squares 2 units away from the starting square 1 that also contain the digit 1 - 11, 15, and 19. Of these 11 is the smallest so is the square stepped to.
a(3) = 10. Of the two adjacent squares to 11 that also contain the digit 1 the square 10 is the smallest.
a(4) = 12. This is the only unvisited square within 2 units of a(3) = 10 that also contains the digit 1.
a(12) = 39. This is the only unvisited square within sqrt(2) units of a(11) = 19 that contains either the digit 3 or 9. It is also the first square stepped to that does not share the digit 1 with the previous square.
Squares visited by a chess rook moving on a square-spiral numbered board where the rook moves to the closest unvisited square containing a prime number. In case of a tie it chooses the square with the smallest prime number.
+10
5
1, 2, 3, 5, 7, 41, 43, 109, 107, 103, 37, 193, 191, 97, 101, 199, 197, 683, 677, 673, 1753, 1747, 1429, 1427, 887, 883, 661, 659, 881, 877, 307, 461, 463, 653, 1129, 1733, 2083, 2081, 3323, 3319, 3797, 3793, 5419, 5417, 5413, 4297, 2861, 2857, 2447, 2069, 1723, 1721, 1409, 1123, 1117, 1399
COMMENTS
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.
LINKS
Scott R. Shannon, Image showing the 350 steps of the rook's path. A green square shows the starting 1 square, a red square shows the final square with number 2179, 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. The four squares which block the rook's movement from the final square are shown with a red border. The square spiral numbering of the board is shown as a thin white line. Click on the image to zoom in to see the prime numbers.
EXAMPLE
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) = 1, the starting square for the rook.
a(2) = 2. The four unvisited prime numbered squares around a(1) the rook can move to are numbered 2,61,19,23. Of these 2 is the closest, being 1 unit away.
a(3) = 3. The three unvisited prime numbered squares around a(2) = 2 the rook can move to are numbered 47,11,3. Both 11 and 3 are 1 units away, and of those 3 is the smallest.
a(7) = 43. The three unvisited prime numbered squares around a(6) = 41 the rook can move to are numbered 37,43,107. Both 43 and 107 are 2 units away, and of those 43 is the smallest. Note that 37, the smallest available prime, is 4 units away.
a(230) = 8887. There is only one unvisited prime numbered square around a(229) = 2143 the rook can move to. The square 8887 is 24 units away to the left of 2143.
CROSSREFS
Cf. A336402, A336446, A336447, A330979, A000040, A063826, A214664, A214665, A136626, A115258, A331027.
Squares visited by a chess rook moving on a square-spiral numbered board where the rook moves to an unvisited square containing the smallest prime number.
+10
5
1, 2, 3, 5, 7, 41, 37, 31, 29, 521, 509, 337, 109, 43, 47, 83, 89, 179, 173, 359, 353, 349, 113, 293, 307, 311, 313, 317, 191, 97, 101, 103, 107, 691, 683, 197, 193, 1429, 1427, 887, 883, 661, 659, 653, 463, 461, 457, 181, 467, 479, 1163, 1171, 331, 673, 677, 1153, 1151, 487, 491, 199
COMMENTS
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.
LINKS
Scott R. Shannon, Image showing the 134 steps of the rook's path. A green square shows the starting 1 square, a red square shows the final square with number 863, 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. The three squares which block the rook's movement from the final square are shown with a red border. The square spiral numbering of the board is shown as a thin white line. Click on the image to zoom in to see the prime numbers.
EXAMPLE
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) = 1, the starting square for the rook.
a(2) = 2. The four unvisited prime numbered squares around a(1) the rook can move to are numbered 2,61,19,23. Of these 2 is the smallest.
a(7) = 37. The three unvisited prime numbered squares around a(6) = 41 the rook can move to are numbered 37,43,107. Of those 37 is the smallest. Note that 43 is the closest prime, being only 2 units away while 37 is 4 units away.
a(135) = 863. The final square. The three previously visited prime numbered squares around a(135) are numbered 191,859,1709. Note there is no fourth prime as the column of squares directly upward from 863 contains no primes; the values from 871,994,1125,... and beyond fit the quadratic 4n^2+119n+871, which can be factored as (4n+67)*(n+13), and thus contains no primes.
CROSSREFS
Cf. A336446, A336402, A336413, A330979, A000040, A063826, A214664, A214665, A136626, A115258, A331027.
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.
+10
5
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
COMMENTS
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.
LINKS
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.
EXAMPLE
The square spiral is numbered as follows:
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17--16--15--14--13 .
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18 5---4---3 12 29
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19 6 1---2 11 28
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20 7---8---9--10 27
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21--22--23--24--25--26
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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.
Squares visited by a chess queen moving on a square-spiral numbered board where the queen moves to the closest unvisited square containing a prime number. In case of a tie it chooses the square with the smallest prime number.
+10
4
1, 2, 3, 11, 29, 13, 31, 59, 61, 97, 139, 191, 251, 193, 101, 103, 67, 37, 17, 5, 19, 7, 23, 47, 79, 163, 281, 353, 283, 433, 521, 617, 523, 619, 439, 359, 223, 167, 227, 293, 229, 173, 83, 233, 127, 53, 179, 131, 89, 137, 389, 313, 311, 467, 383, 307, 241, 239, 181, 457, 547, 643
COMMENTS
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.
LINKS
Scott R. Shannon, Image showing the 519 steps of the queen's path. A green square shows the starting 1 square, a red square shows the final square with number 1289, 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. The eight squares which block the queen's movement from the final square are shown with a red border. The square spiral numbering of the board is shown as a thin white line. Click on the image to zoom in to see the prime numbers.
EXAMPLE
The board is numbered with the square spiral:
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17--16--15--14--13 .
| | .
18 5---4---3 12 29
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19 6 1---2 11 28
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20 7---8---9--10 27
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21--22--23--24--25--26
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a(1) = 1, the starting square for the queen.
a(2) = 2. The seven unvisited prime numbered squares around a(1) the queen can move to are numbered 2,3,61,5,19,7,23. Of these 2 is the closest, being 1 unit away. There are no primes in the south-east direction from a(1).
a(4) = 11. The four unvisited prime numbered squares around a(3) = 3 the queen can move to are numbered 11,29,13,5, the other two directions not having any primes. Both 11 and 13 are sqrt(2) units away, and of those 11 is the smallest.
a(40) = 227. The three unvisited prime numbered squares around a(39) = 167 the queen can move to are numbered 227,173,53, Of these 227 is the closest, being 4 units away. Note that the square with prime number 83 is only sqrt(10), about 3.16, units away but is at relative coordinates (1,3) to 167 so cannot be reach by the queen.
CROSSREFS
Cf. A336413, A336446, A336447, A330979, A000040, A063826, A214664, A214665, A136626, A115258, A331027.
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