A357985 -id:A357985 - cp's OEIS frontend

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.

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.

A363824 a(0) = 0; for n > 0, a(n) is the total number of other numbers, being constructed on a square spiral, that are visible from a(n-1) that equal a(n-1).

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 2, 0, 3, 0, 3, 1, 1, 2, 2, 3, 2, 2, 5, 0, 3, 2, 4, 0, 5, 0, 4, 1, 3, 3, 4, 1, 3, 4, 1, 3, 5, 1, 4, 1, 5, 2, 5, 2, 6, 0, 4, 2, 7, 0, 4, 5, 4, 4, 5, 3, 6, 1, 5, 3, 5, 4, 9, 0, 4, 6, 1, 8, 0, 9, 1, 7, 0, 10, 0, 10, 0, 9, 2, 7, 2, 6, 3, 6, 3, 6, 1, 7, 2, 9, 1, 7, 3, 6, 4, 7, 2, 8, 0

Offset: 0

Scott R. Shannon, Oct 19 2023

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

			The spiral begins:
.
                                .
    5---3---1---4---3---1---4   :
    |                       |   :
    1   2---3---2---2---1   3   5
    |   |               |   |   |
    4   2   2---0---1   1   3   4
    |   |   |       |   |   |   |
    1   5   0   0---0   3   1   4
    |   |   |           |   |   |
    5   0   2---0---3---0   4   5
    |   |                   |   |
    2   3---2---4---0---5---0   4
    |                           |
    5---2---6---0---4---2---7---0
.
a(1) = 0 as a(0) = 0, and there are currently no other numbers that equal 0.
a(2) = 1 as a(1) = 0, and from a(1), at (1,0) relative to the starting square, there is currently one other visible 0, namely a(0).
a(6) = 2 as a(5) = 0, and from a(5), at (-1,0) relative to the starting square, there are currently two other visible 0's, namely a(0) and a(3). Note that a(1) = 0 is not visible as it is hidden by a(0).

Scott R. Shannon, Table of n, a(n) for n = 0..10000
Scott R. Shannon, Image of the first 500000 terms.
Scott R. Shannon, Image of the first 10000 terms on the square spiral The colors are graduated across the spectrum to show their relative size. Zoom in to see the numbers.

Cf. A331400, A357991, A357985, A347358, A347357.

cp's OEIS Frontend

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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