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.

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

0,5

Comments

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.

Links

Table of n, a(n) for n=0..54.

Example

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.

Crossrefs

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

Sequence in context: A220377 A329696 A145969 * A140352 A277130 A082588

Adjacent sequences: A357982 A357983 A357984 * A357986 A357987 A357988

Keyword

sign

Author

Scott R. Shannon, Oct 23 2022

Status

approved