A348025 The numbers visited on a square spiral when stepping to the smallest unvisited number that is not visible from and shares a divisor > 1 with the current number. Start with 1 and 11.

1, 11, 55, 15, 21, 3, 9, 27, 30, 2, 6, 14, 16, 10, 4, 8, 12, 18, 20, 32, 34, 28, 22, 24, 26, 36, 33, 39, 42, 38, 40, 46, 44, 48, 45, 5, 25, 65, 13, 91, 7, 35, 49, 105, 51, 17, 153, 57, 19, 114, 50, 52, 54, 56, 66, 68, 62, 58, 29, 87, 84, 60, 63, 69, 23, 161, 77, 99, 93, 31, 124, 70, 72

Offset

1,2

Comments

A number is not visible from the current number if, given it has coordinates (x,y) relative to the current number, the greatest common divisor of |x| and |y| is greater than 1. The sequence has a(2) = 11 as that is the smallest number not visible from a(1) = 1.

In the first 10000 terms the longest single step is one at n = 6888 of length sqrt(22556) units between 22203 to 7389. The maximum difference between terms in the same range is from 3469 to 58973 at n = 9709.

Links

Table of n, a(n) for n=1..73.

Scott R. Shannon, Image of the path for the first 10000 terms. The colors are graduated across the spectrum to show the relative step order.

Example

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) = 55 as gcd(55,11) = 11 and 55 is unvisited and not visible from 11, being blocked by 29.
a(4) = 15 as gcd(15,55) = 5 and 15 is unvisited and not visible from 55, being blocked by 13, 14 and 30.
a(5) = 21 as gcd(21,15) = 3 and 21 is unvisited and not visible from 15, being blocked by 6.

Crossrefs

Cf. A348022 (visible), A331400, A335661, A063826, A332767.

Sequence in context: A061983 A079884 A200172 * A050900 A246406 A255415

Adjacent sequences: A348022 A348023 A348024 * A348026 A348027 A348028

Keyword

nonn

Author

Scott R. Shannon, Sep 25 2021

Status

approved