A355270 -id:A355270 - cp's OEIS frontend

A355271 Lexicographically earliest sequence of positive integers on a square spiral such that the product of adjacent pairs of numbers within each row, column and diagonal is distinct in that row, column and diagonal.

1, 1, 1, 1, 2, 2, 3, 2, 4, 3, 3, 4, 2, 3, 4, 4, 5, 3, 2, 5, 4, 3, 5, 4, 2, 2, 3, 5, 2, 2, 4, 2, 3, 5, 4, 6, 3, 1, 1, 5, 5, 4, 1, 1, 6, 6, 2, 5, 6, 4, 5, 1, 1, 6, 4, 7, 5, 4, 1, 5, 3, 6, 2, 3, 1, 1, 3, 7, 6, 2, 7, 4, 5, 7, 3, 6, 1, 1, 4, 3, 1, 5, 2, 1, 1, 6, 5, 7, 1, 5, 3, 3, 5, 1, 1, 3, 7, 4, 6

Offset: 1

Scott R. Shannon, Jun 26 2022

nonn

In the first 2 million terms the largest number is 257, while the number 37, the most commonly occurring number, appears 43477 times. Prime numbers appear more often than the composites. See the linked images.

			The spiral begins:
.
                                .
    3---6---4---5---3---2---4   :
    |                       |   :
    1   5---4---4---3---2   2   4
    |   |               |   |   |
    1   3   2---1---1   4   2   6
    |   |   |       |   |   |   |
    5   2   2   1---1   3   5   1
    |   |   |           |   |   |
    5   5   3---2---4---3   3   1
    |   |                   |   |
    4   4---3---5---4---2---2   5
    |                           |
    1---1---6---6---2---5---6---4
.
a(25) = 2 as when a(25) is placed, at coordinate (2,-2) relative to the starting square, its adjacent squares are a(10) = 3, a(9) = 4, a(24) = 4. The products of adjacent pairs of numbers in a(25)'s column are 3 * 3 = 9, 3 * 4 = 12, 4 * 2 = 8, in its north-west diagonal are 4 * 1 = 4, 1 * 2 = 2, 2 * 5 = 10, and in its row are 4 * 5 = 20, 5 * 3 = 15, 3 * 4 = 12. Setting a(25) to 1 would create a product of 4 with its diagonal neighbor 4, but 4 has already occurred as a product on this diagonal. Similarly numbers 3, 4 and 5 would not be possible as they would create products with the three adjacent numbers, 3, 4, and 4, which have already occurred along the corresponding column, diagonal or row. But 2 is smaller and creates new products, namely 6, 8 and 8, with its three neighbors that have not already occurred along the corresponding column, diagonal and row.

Scott R. Shannon, Image of the first 2 million 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.
Scott R. Shannon, Distribution of a(n) for the first 2 million terms. The number 37, the most commonly occurring number, appears 43477 times. The prime numbers are shown in yellow, nonprimes in white.

Cf. A355270, A274640, A275609, A307834.

A357985 Counterclockwise square spiral constructed using the integers so that a(n) plus all other numbers currently visible from the current number equals n; start with a(0) = 0.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 3, -1, 6, -2, -1, 0, 1, 9, -8, 15, -5, -7, -10, 14, -29, 58, -78, 101, -118, 150, -61, 309, -307, 553, -494, -186, -644, 315, -1177, 731, -1458, 3480, -5183, 7096, -8328, 9735, -10882, 7200, -29452, 31322, -52670, 51401, -65210, 61001, 11318, 135012, -109687, 259226, -221542

Offset: 0

Scott R. Shannon, Oct 23 2022

sign

A number is visible from the current number if, given that it has coordinates (x,y) relative to the current number, the greatest common divisor of |x| and |y| is 1.

The magnitude of the numbers grow surprisingly quickly, e.g., a(150) = -4346232663618226. The only known terms that equal zero are a(0) and a(11); it is unknown whether more exist or if all integers eventually appear.

			The spiral begins:
.
                                 .
                                 .
    -5....15...-8....9.....1    553
     |                     |     |
    -7    2....1.....1     0   -307
     |    |          |     |     |
   -10    1    0.....1    -1    309
     |    |                |     |
    14    3...-1.....6... -2    -61
     |                           |
   -29...58...-78...101...-118...150
.
.
a(6) = 3 as from square 6, at (-1,1) relative to the starting square, the numbers currently visible are 1 (at -1,0), 0 (at 0,0), 1 (at 0,1), and 1 (at 1,0). These four numbers sum to 3, so a(6) = 3 so that 3 + 3 = 6.
a(7) = -1 as from square 7, at (0,-1) relative to the starting square, the numbers currently visible are 3 (at -1,-1), 1 (at -1,0), 2 (at -1,1), 0 (at 0,0), 1 (at 1,1), and 1 (at 1,0). These six numbers sum to 8, so a(7) = -1 so that -1 + 8 = 7.

Cf. A357991, A307834, A275609, A274640, A355270.

A357991 Lexicographically earliest counterclockwise square spiral constructed using the nonnegative integers so that a(n) plus all other numbers currently visible from the current number form a distinct sum; start with a(0) = 0.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 3, 0, 4, 0, 0, 0, 1, 5, 0, 6, 0, 0, 1, 0, 2, 4, 0, 7, 0, 8, 0, 7, 0, 7, 0, 0, 0, 0, 0, 0, 0, 12, 0, 13, 0, 16, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 12, 0, 22, 0, 19, 0, 20, 1, 0, 0, 0, 0, 0, 0, 0, 0, 17, 0, 25, 0, 24, 0, 20, 1, 26, 0, 28, 0, 26, 0, 31, 0, 31, 0, 0, 0, 0

Offset: 0

Scott R. Shannon, Oct 23 2022

nonn

A number is visible from the current number if, given that it has coordinates (x,y) relative to the current number, the greatest common divisor of |x| and |y| is 1.

In the first 50000 terms the smallest number that has not appeared is 9; it is unknown if all the positive numbers eventually appear.

			The spiral begins:
.
                       .
                       .
   0---6---0---5---1   7
   |               |   |
   0   2---1---1   0   0
   |   |       |   |   |
   1   1   0---1   0   7
   |   |           |   |
   0   3---0---4---0   0
   |                   |
   2---4---0---7---0---8
.
.
a(6) = 3 as from square 6, at (-1,-1) relative to the starting square, the numbers currently visible are 1 (at -1,0), 0 (at 0,0), 1 (at 1,0), and 1 (at 0,1). These three numbers sum to 3, so a(6) = 3 so that 3 + 3 = 6, the smallest sum that has not previous occurred.
a(8) = 4 as from square 8, at (1,-1) relative to the starting square, the numbers currently visible are 0 (at 0,-1), 1 (at -1,0), 0 (at 0,0), 1 (at 1,0), and 1 (at 0,1). These five numbers sum to 3, so a(8) = 4 so that 3 + 4 = 7, the smallest sum that has not previous occurred. Note that a(7) = 0 and forms a sum of 8.

Cf. A357985, A307834, A275609, A274640, A355270.

A361724 Lexicographically earliest sequence of distinct positive numbers on a square spiral such that the eight sums of each number with its eight nearest neighbors are distinct across the entire spiral and no number on the spiral equals any such sum.

Original entry on oeis.org

1, 2, 4, 7, 12, 14, 16, 22, 27, 10, 31, 40, 39, 46, 47, 20, 45, 52, 61, 60, 18, 80, 68, 81, 82, 70, 89, 94, 83, 48, 62, 105, 100, 69, 117, 25, 111, 129, 127, 124, 143, 106, 112, 132, 155, 119, 126, 128, 63, 56, 157, 158, 107, 178, 193, 168, 118, 170, 55, 195, 189, 197, 192, 206, 182, 211, 202

Offset: 1

Scott R. Shannon and Eric Angelini, Mar 22 2023

			a(3) = 4 as a(1) + a(2) = 1 + 2 = 3, so a(3) cannot 1,2 or 3. a(3) has a(1) = 1 and a(2) = 2 as neighbors which form sums 4 + 1 = 5 and 4 + 2 = 6 neither of which have appeared, so 4 can be chosen.
a(5) = 12 as the numbers already used are 1,2,4,7, which form the sums 3,5,8,6,9,11 with their nearest neighbors. The lowest free number is therefore 10, but a(5) has a(1) = 1 as a neighbor and would create the sum 10 + 1 = 11 which has already appeared as a sum. The next free number is 12 which forms sums 12 + 7 = 19 and 12 + 1 = 13 which have not appeared, so 12 can be chosen.

Scott R. Shannon, Image of the first 50000 terms. The green line is a(n) = n.
Scott R. Shannon, Image of the first 50000 terms on the square spiral. The numbers are spread across the spectrum from red to violet to show their relative size. Zoom in to see the numbers.
Scott R. Shannon, Image of the first 200000 terms.

Cf. A274640, A355270, A358151, A307834, A174344, A274923

A355314 Lexicographically earliest sequence of positive integers on a square spiral such that the difference between all orthogonally adjacent pairs of numbers is distinct.

Original entry on oeis.org

0, 0, 1, 3, 7, 12, 1, 7, 15, 1, 10, 23, 0, 17, 35, 54, 0, 27, 48, 72, 0, 26, 55, 83, 31, 0, 34, 69, 106, 39, 1, 41, 83, 126, 1, 45, 91, 140, 77, 128, 2, 57, 1, 61, 119, 183, 1, 93, 158, 1, 74, 143, 218, 0, 115, 192, 0, 79, 160, 244, 2, 87, 174, 1, 89, 185, 1, 166, 6, 101, 198, 296, 0, 101, 203, 1

Offset: 0

Scott R. Shannon, Jun 28 2022

nonn

For larger n the sequences typically consists of a repeating pattern of three values - the first one is small, less than 5, a second larger value, and then a third even larger value, typically around double the previous value. However this pattern is occasionally broken by a fourth or fifth larger value which shifts the position of the subsequent repeating block of three values. This leads to the overall spiral pattern showing a uniform pattern of numbers crossed by random zig-zag lines of values not following the three-value pattern. See the linked color image.

			The spiral begins:
.
                                .
   91--45---1--126-83--41---1   :
    |                       |   :
   140  0--54--35--17---0  39  115
    |   |               |   |   |
   77  27   7---3---1  23  106  0
    |   |   |       |   |   |   |
   128 48  12   0---0  10  69  218
    |   |   |           |   |   |
    2  72   1---7--15---1  34  143
    |   |                   |   |
   57   0--26--55--83--31---0  74
    |                           |
    1--61--119-183--1--93--158--1
.
.
a(8) = 15 as when a(8) is placed, at coordinate (1,-1) relative to the starting square, its two orthogonally adjacent squares are a(1) = 0 and a(7) = 7. The ten previously occurring differences between all orthogonally adjacent pairs up to a(7) are 0, 1, 2, 3, 4, 5, 6, 7, 11, 12. The lowest unused difference is 8 thus a(8) = 15 can be chosen as it results in differences with its two orthogonal neighbors of 15 - 7 = 8 and 15 = 0 = 15, neither of which has previously occurred.

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. A307834, A275609, A274640, A355270.

A363665 Starting with a(1) = 1, the lexicographically earliest sequence of integers on a square spiral such that every number equals the sum of its eight adjacent neighbors.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -2, 0, 0, 1, -2, 0, 0, 0, -1, 0, 0, 0, -2, 2, 0, 0, -1, 2, -3, 0, 0, 3, -2, 2, -3, 0, 0, 3, -1, -1, 1, 0, 0, 1, 0, 0, -5, 8, 0, 0, -5, 4, 0, 4, -7, 0, 0, 6, -6, 4, -4, 8, -7, 0, 0, 2, -2, 4, -4, 0, 7, 0, 0, -8, 6, 3, -8, 10, -15, 16, 0, 0, -9, 6, -5, 7, -8, 13

Offset: 1

Scott R. Shannon, Jun 14 2023

sign

As the terms are not distinct the first two numbers of any new row or column will always be zero. In the first 500000 terms the last zero that is not at the beginning of a row or column is a(190) = 0. Is it unknown if more such zeros exist. In the same range the smallest positive numbers not yet occurring are 5, 9, 11, 12, 15, 19, 20, ... . It is unknown if all integers eventually appear. The terms increase rapidly in size; in the first 500000 terms the largest positive term is a(499848) = 1267...5398, a number with 226 digits.

			The spiral begins:
.                               .
.                               |
    0__-3___2__-2___3___0___0  -7
    |                       |   |
    0   0__-2___1___0___0  -3   4
    |   |               |   |   |
    3   0   0___0___0  -2   2   0
    |   |   |       |   |   |   |
   -1   0   0   1___0   0  -1   4
    |   |   |           |   |   |
   -1  -1   0___0___1___0   0  -5
    |   |                   |   |
    1   0___0___0__-2___2___0   0
    |                           |
    0___0___1___0___0__-5___8___0
.
.
a(9) = 1 as a(1) = 1 and a(2)..a(8) = 0, therefore a(9) = 1 so the sum of the eight numbers around a(1) equals 1.
a(12) = -2 as a(2) = 0 while a(1), a(9) = 1, a(2)..a(4), a(8), a(10), a(11) = 0, therefore a(12) = -2 so the sum of the eight numbers around a(1) equals 0.

Scott R. Shannon, Table of n, a(n) for n = 1..10000.
Scott R. Shannon, Image of log_10(a(n)+1) of the absolute value of the first 500000 terms.
Scott R. Shannon, Image showing the sign of the first 500000 terms on the spiral. White is positive, black is negative, yellow is zero.

Cf. A358151, A358337, A361724, A355270.

cp's OEIS Frontend

A355271 Lexicographically earliest sequence of positive integers on a square spiral such that the product of adjacent pairs of numbers within each row, column and diagonal is distinct in that row, column and diagonal.

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A357985 Counterclockwise square spiral constructed using the integers so that a(n) plus all other numbers currently visible from the current number equals n; start with a(0) = 0.

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A357991 Lexicographically earliest counterclockwise square spiral constructed using the nonnegative integers so that a(n) plus all other numbers currently visible from the current number form a distinct sum; start with a(0) = 0.

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A361724 Lexicographically earliest sequence of distinct positive numbers on a square spiral such that the eight sums of each number with its eight nearest neighbors are distinct across the entire spiral and no number on the spiral equals any such sum.

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A355314 Lexicographically earliest sequence of positive integers on a square spiral such that the difference between all orthogonally adjacent pairs of numbers is distinct.

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A363665 Starting with a(1) = 1, the lexicographically earliest sequence of integers on a square spiral such that every number equals the sum of its eight adjacent neighbors.

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