A332583 Label only the prime-numbered position cells of the infinite 2D square lattice with the square spiral (or Ulam spiral), starting with 1 at the center; sequence lists primes that are visible from square 1.

2, 3, 5, 7, 19, 23, 29, 41, 47, 59, 61, 67, 71, 79, 83, 89, 97, 103, 107, 109, 113, 131, 137, 149, 167, 173, 179, 181, 191, 193, 199, 223, 227, 229, 239, 251, 263, 271, 277, 283, 293, 311, 317, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 409, 419, 433, 439, 443, 449, 457, 461, 467, 479, 487, 491, 499, 503

Offset

1,1

Comments

Any grid point labeled with a prime number and with coordinates (x,y) relative to the central grid point, which is numbered 1, and where the greatest common divisor (gcd) of |x| and |y| equals 1 will be visible from the central point. Grid points where gcd(|x|,|y|) > 1 may have another prime grid point directly between it and the central point and will thus not be visible.

For a square spiral of size 10001 by 10001, slightly over 100 million numbers, a total of 5762536 primes are present, of which 4811013 are visible. This gives a ratio of visible primes to all primes of about 0.835.

Links

Table of n, a(n) for n=1..67.

Scott R. Shannon, Image showing the visible primes from point 1 for the first 100000 grid points. The primes visible from the central 1 square are shown in yellow while those blocked are shown in grey. The blocked primes also contain the number in parenthesis of the prime which blocks their visibility from the central square. Zoom into the image to see the grid point numbers.

Eric Weisstein's World of Mathematics, Visible Point.

Wikipedia, Ulam Spiral.

Example

The 2D grid is shown below. The primes that are blocked from the central 1 square are in parentheses; these all have another prime number directly between their position and the central square.
.
.
-------------61-------59------+
                              |
(37)---------------------(31) |
|                         |   |
|  (17)--------------(13) |   |
|    |                |   |   |
|    |   5--------3   |   29  |
|    |   |        |   |   |   |
|   19   |   1----2  (11) | (53)
|    |   |            |   |   |
41   |   7------------+   |   |
|    |                    |   |
|    +-------23-----------+   |
|                             |
(43)-------------47-----------+
.
.
a(1) = 2 to a(4) = 7 are all primes adjacent to the central 1 point, thus all are visible from that square.
a(5) = 19 as primes 11,13,17 are blocked from the central 1 point by points with prime numbers 2,3,5 respectively.
a(14) = 79 as although the point 79 has relative coordinates of (2,-4) from the central square, gcd(|2|,|-4|) = 2, there is no other prime at coordinate (1,-2), thus it is visible. This square is not visible from the central square when nonprime points are also considered in the spiral.

Crossrefs

Cf. A000040, A063826, A157426, A157428, A325604, A325606, A331400, A332582.

Cf. also A156859.

Sequence in context: A360041 A360040 A244529 * A345335 A025019 A140327

Adjacent sequences: A332580 A332581 A332582 * A332584 A332585 A332586

Keyword

nonn

Author

Scott R. Shannon, Feb 17 2020

Extensions

Edited by N. J. A. Sloane, Feb 17 2020

Status

approved