A332582 Label the 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, 29, 41, 47, 83, 89, 97, 103, 107, 109, 113, 173, 179, 181, 191, 193, 199, 223, 293, 311, 317, 347, 353, 359, 443, 449, 457, 461, 467, 479, 487, 491, 499, 503, 509, 521, 523, 631, 641, 643, 647, 653, 659, 661, 673, 683, 691, 701, 709, 719, 727, 857, 863, 887, 929, 947, 953, 1091

Offset

1,1

Comments

Any grid point with relative coordinates (x,y) from 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 will have another point directly between it and the central point and will thus not be visible. In an infinite 2D square lattice the ratio of visible grid points to all points is 6/Pi^2, approximately 0.608, the same as the probability of two random numbers being relative prime.

For a square spiral of size 10001 X 10001, slightly over 100 million numbers, a total of 60803664 numbers are visible, of which 2155170 are prime. The total number of primes in the same range is 5762536, giving a ratio of visible primes to all primes of about 0.374. This is significantly lower than the ratio for all numbers of 0.608, indicating a prime is more likely to be hidden from the origin than a random number.

Primes p such that A174344(p) and A268038(p) are coprime. - Robert Israel, Feb 16 2024

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Scott R. Shannon, Image showing the visible primes from point 1 for the first 100000 grid points. The visible primes are highlighted in yellow while the hidden primes are highlighted in red. The central 1 square is white while the nonprimes are gray. 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. Composite numbers are shown as a '*'. The primes that are blocked from the central 1 square are in parentheses; these all have another composite or 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) = 29 as primes 11, 13, 17, 19, 23 are blocked from the central 1 point by points numbered 2, 3, 5, 6, 8 respectively.

Maple

x:= 0: y:= 0: R:= NULL: count:= 0:
for i from 2 while count < 100 do
  if x >= y then
    if x < -y + 1 then x:= x+1
    elif x > y then y:= y+1
    else x:= x-1
    fi
  elif x <= -y then y:= y-1
    else x:= x-1
  fi;
  if isprime(i) and igcd(abs(x), abs(y))=1 then R:= R, i; count:= count+1 fi
od:
R; # Robert Israel, Feb 16 2024

Crossrefs

Cf. A000040, A063826, A157426, A157428, A174344, A268038, A325604, A325606, A331400, A332583.

Sequence in context: A215155 A228486 A117135 * A019372 A117299 A091924

Adjacent sequences: A332579 A332580 A332581 * A332583 A332584 A332585

Keyword

nonn

Author

Scott R. Shannon, Feb 17 2020

Status

approved