A344367 -id:A344367 - cp's OEIS frontend

A358048 Lexicographically earliest sequence of distinct nonnegative integers on a square spiral such that every number shares a digit with each of its eight surrounding neighbors.

0, 10, 20, 30, 40, 50, 60, 70, 80, 18, 100, 12, 2, 23, 102, 90, 49, 104, 101, 103, 16, 105, 106, 107, 78, 8, 81, 1, 21, 22, 24, 25, 26, 29, 19, 9, 39, 91, 14, 11, 13, 15, 17, 31, 41, 51, 71, 87, 28, 38, 48, 108, 61, 112, 27, 32, 34, 42, 52, 62, 69, 59, 79, 89, 83, 93, 94, 109, 110, 111, 113

Offset: 0

Eric Angelini and Scott R. Shannon, Oct 27 2022

The sequence is conjectured to be a permutation of the nonnegative integers.

			The square spiral begins:
                           .
                           .
   49--90--102---23---2   22
   |                  |    |
  104  40---30---20   12  21
   |   |          |   |    |
  101  50    0---10  100   1
   |   |              |    |
  103  60---70---80---18  81
   |                       |
   16--105--106--107--78---8
.
a(11) = 12 as when the eleventh square is filled its surrounding neighbors are 10, 20, 100, and 12 is the smallest unused number that shares a digit with each of these three numbers.

Eric Angelini, More inspiring spirals (1), personal blog CinquanteSignes.blogspot.com, Oct. 23, 2022.
Gleb Ivanov, Python program.

Cf. A358021, A344325, A344367, A354111, A343530.

A358021 Lexicographically earliest sequence of distinct nonnegative integers on a square spiral such that no number shares a digit with any of its eight surrounding neighbors.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 30, 44, 10, 55, 11, 20, 13, 66, 12, 33, 14, 22, 15, 23, 16, 24, 57, 18, 25, 36, 27, 48, 26, 34, 56, 47, 28, 50, 49, 58, 60, 29, 35, 67, 38, 40, 77, 59, 37, 19, 68, 39, 46, 70, 41, 80, 31, 65, 90, 17, 88, 21, 93, 51, 43, 69, 71, 32, 111, 73, 81, 64, 72, 89, 42, 91

Offset: 0

Scott R. Shannon and Eric Angelini, Oct 24 2022

The sequence is finite; after a(264) = 177 the square at coordinate (-8,-1) relative to the starting square is reached which has the four neighbors 177, 409, 256, 308. These numbers include all digits 0 to 9 so the next term does not exist. The largest used number is a(241) = 7777 while smallest unused number is 120.

			The square spiral begins:
                       .
                       .
  13--20--11--55--10  36
   |               |   |
  66   4---3---2  44  25
   |   |       |   |   |
  12   5   0---1  30  18
   |   |           |   |
  33   6---7---8---9  57
   |                   |
  14--22--15--23--16--24
.
a(10) = 30 as when the tenth square is filled its surrounding neighbors are 1, 2, 8, 9, and 30 is the smallest unused number that does not contain those digits.

Cf. A358048, A344325, A344367, A354111, A343530.

A358254 Lexicographically earliest sequence of distinct nonnegative integers on a square spiral such that the sum of the eight numbers around any chosen number ends in the chosen number.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 12, 8, 9, 10, 11, 13, 15, 23, 14, 16, 18, 21, 17, 19, 29, 25, 33, 20, 22, 26, 28, 120, 24, 27, 87, 58, 125, 88, 30, 31, 97, 124, 45, 187, 32, 34, 73, 132, 55, 49, 42, 35, 36, 95, 195, 59, 98, 863, 37, 38, 130, 104, 129, 62, 736, 67, 39, 40, 115, 131, 48, 748, 82, 208, 41

Offset: 0

Eric Angelini and Scott R. Shannon, Nov 05 2022

The first two numbers placed when a new row or column is formed around the spiral do not complete a 3x3 block of numbers, thus they can be the lowest two numbers that have not appeared. Therefore the sequence a permutation of the nonnegative integers.

In the first 10000 terms the largest value is a(8526) = 9874588 while the smallest unused number is 902.

			The square spiral begins:
                       .
                       .
  14--23--15--13--11  120
   |               |   |
  16   4---3---2  10  28
   |   |       |   |   |
  18   5   0---1   9  26
   |   |           |   |
  21   6---7--12---8  22
   |                   |
  17--19--29--25--33--20
.
a(8) = 12 as when the ninth cell is filled it completes a ring of eight numbers around the central cell with number 0, therefore the sum of these eight numbers must end in 0. The sum around the central cell when the eighth cell is filled is 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28, and the lowest unused number that can be added so that the sum ends in 0 is 28 + 12 = 40, so a(8) = 12.
a(29) = 120 as when the thirtieth cell is filled the sum of the previous numbers around the number 10 is 13 + 11 + 2 + 28 + 1 + 9 + 26 = 90, and since 20 has already appeared the smallest unused number that can be added to 90 to form a number that ends in 10 is 120.

Scott R. Shannon, Table of n, a(n) for n = 0..5000
Eric Angelini, 9 frames and 1 super-frame, personal blog CinquanteSignes.blogspot.com, Oct. 30, 2022.

Cf. A358048, A358021, A358021, A344325, A344367, A354111, A343530.

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.

A358129 Lexicographically earliest sequence of distinct nonnegative integers on a square spiral such that no number shares a digit with any of its four orthogonally adjacent neighbors.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 20, 11, 22, 10, 24, 13, 25, 16, 23, 14, 26, 15, 28, 17, 30, 12, 33, 18, 27, 19, 32, 40, 29, 31, 42, 36, 21, 34, 50, 41, 35, 44, 37, 45, 38, 46, 39, 47, 53, 48, 55, 49, 52, 43, 56, 70, 51, 60, 57, 61, 54, 63, 58, 64, 59, 66, 71, 62, 73, 68, 72, 65, 74, 69, 77, 90, 75

Offset: 0

Scott R. Shannon and Eric Angelini, Oct 30 2022

This sequence is similar to A358021 except here only the four orthogonally adjacent neighbors are considered. It is unknown if the sequence eventually terminates; if it does, it does so after more than 1.5 million terms. In that range on numerous occasions the only available number is one containing only the digit 7. For example a(862833) = 7...7, where there are forty-nine 7's.

			The square spiral begins:
                       .
                       .
  25--13--24--10--22  19
   |               |   |
  16   4---3---2  11  27
   |   |       |   |   |
  23   5   0---1  20  18
   |   |           |   |
  14   6---7---8---9  33
   |                   |
  26--15--28--17--30--12
.
a(10) = 20 as when the tenth cell is filled its two orthogonal neighbors use the digits 1 and 9, and 20 is the smallest unused number that does not contain either of those digits.

Scott R. Shannon, Image of the first 500000 terms. The values are scaled across the spectrum from red to violet, with the value ranges increasing towards the violet end to give more color weighting to the larger numbers.

Cf. A358021, A358048, A344325, A344367, A354111, A343530.

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A358048 Lexicographically earliest sequence of distinct nonnegative integers on a square spiral such that every number shares a digit with each of its eight surrounding neighbors.

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A358021 Lexicographically earliest sequence of distinct nonnegative integers on a square spiral such that no number shares a digit with any of its eight surrounding neighbors.

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A358254 Lexicographically earliest sequence of distinct nonnegative integers on a square spiral such that the sum of the eight numbers around any chosen number ends in the chosen number.

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A358129 Lexicographically earliest sequence of distinct nonnegative integers on a square spiral such that no number shares a digit with any of its four orthogonally adjacent neighbors.

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