A330979 -id:A330979 - cp's OEIS frontend

A330782 The records for distance squared for step lengths between adjacent composite numbers in A332767, the visited composite numbers for a walk stepping to the closest unvisited composite number on the 2D square (Ulam) spiral.

1, 2, 8, 32, 40, 68, 98, 148, 162, 356, 450

Offset: 1

Scott R. Shannon, Feb 23 2020

The sequence A332767 gives the visited composite numbers for a walk on the 2D square (Ulam) spiral which starts at 1 and then steps to the square containing the closest unvisited composite number. This sequences lists the records for the square of the step distance between visited composite numbers for that walk. For a walk of 1 million steps the largest square distance is 450, approximately 21.1 units, which occurs between A332767(149464) = 64666, which has coordinates (-127,-22) relative to the starting 1-square, to A332767(149465) = 67774 with coordinates (-130,-43). See A332767 for an image of the walk. It is unknown if this is a finite or infinite sequence.

			The below table shows the details of the record step lengths of this sequence for the first 1 million steps. The coordinate is relative to the starting 1-square.
--------------------------------------------------------------------------------
    a(n)  | A332767 step # |   Start value & coord   |  End value & coord      |
--------------------------------------------------------------------------------
       1  |         1      |         1 (0,0)         |         4 (0,1)         |
       2  |         6      |        32 (2,3)         |        30 (3,2)         |
       8  |       154      |        74 (-3,-4)       |       158 (-5,-6)       |
      32  |      4501      |      5526 (-37,-12)     |      6782 (-41,-16)     |
      40  |     65877      |     48150 (110,79)      |     53558 (116,81)      |
      68  |     91787      |    126154 (178,-49)     |    137780 (186,-47)     |
      98  |    125472      |    145762 (-28,191)     |    156654 (-35,198)     |
     148  |    142733      |    105316 (-147,-162)   |    102746 (-135,-160)   |
     162  |    142741      |     92744 (-129,-152)   |     82106 (-120,-143)   |
     356  |    142869      |     67818 (-130,-87)    |     57792 (-120,-71)    |
     450  |    149464      |     64666 (-127,-22)    |     67774 (-130,-43)    |

Wikipedia, Ulam Spiral.

Cf. A332767, A330979, A331027, A000040, A063826, A136626.

A335585 The numbers visited on a square spiral, with a(n) = n for 1 <= n <= 3, when stepping to an unvisited number as close as possible to the n = 1 starting position that has at least one common factor with the second last visited number but none with the last visited number. In case of a tie, choose the smallest number.

Original entry on oeis.org

1, 2, 3, 4, 9, 8, 15, 14, 5, 6, 25, 12, 35, 16, 7, 10, 21, 20, 27, 22, 39, 11, 18, 77, 24, 49, 34, 63, 17, 28, 51, 40, 33, 46, 45, 23, 30, 161, 26, 69, 13, 36, 65, 32, 55, 38, 75, 19, 42, 95, 44, 85, 48, 115, 52, 105, 62, 87, 68, 29, 54, 203, 60, 119, 76, 153, 70, 117, 50, 57, 56, 81, 58, 93

Offset: 1

Scott R. Shannon, Jan 26 2021

This sequence is the square spiral version of the Yellowstone permutation A098550. The same rules for selecting the next number apply except that, instead of choosing the smallest unvisited number for a(n), the number closest to the starting n = 1 position which satisfies the selection rules is chosen. If two or more such numbers exist then the smallest is chosen.

The first term that differs from A098550 is a(23) = 18. See the examples below.

			The square spiral used is:
.
  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(7) = 15 as a(5) = 9 = 3*3 and a(8) = 8 = 2*2*2, thus a(7) must contain 3 as a factor but not 2. The closest unvisited number to the starting 1 position that satisfies these conditions is 15.
a(23) = 18 as a(21) = 39 = 3*13 and a(22) = 11, thus a(23) must contain 3 or 13 as a factor but not 11. The smallest unvisited number satisfying these conditions is 13, which is sqrt(8) units from 1. However 18 is unvisited and also satisfies the conditions, and is only sqrt(5) units from 1, thus a(23) = 18. This is the first term that differs from A098550.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 50000 visited numbers on the square spiral. The colors are graduated across the spectrum from red to violet to indicate the relative visit order of the numbers. The starting 1 position is colored white.

Cf. A098550, A335661, A330979, A340783, A336957, A064413.

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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A330782 The records for distance squared for step lengths between adjacent composite numbers in A332767, the visited composite numbers for a walk stepping to the closest unvisited composite number on the 2D square (Ulam) spiral.

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