A308412 Indices of Gaussian primes on a square spiral.

3, 5, 7, 9, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 52, 54, 60, 62, 68, 70, 76, 78, 82, 84, 88, 90, 92, 94, 98, 100, 102, 104, 108, 110, 112, 114, 118, 120, 122, 126, 128, 132, 134, 138, 140, 144, 146, 150, 152, 156, 158

Offset: 1

Author

Rémy Sigrist, Jun 01 2019

Keywords

• nonn

Comments

These are the numbers k > 0 such that A174344(k) + i*A274923(k) is a Gaussian prime (where i denotes the imaginary unit).

For symmetry reasons, we obtain the same sequence when considering a clockwise or a counterclockwise square spiral, or when initially moving towards any unit direction.

All terms except the first four are even.

Examples

			The first terms displayed on the center of a counterclockwise square spiral are:
  y\x|    -5   -4   -3   -2   -1    0   +1   +2   +3   +4   +5
  ---+--------------------------------------------------------
   +5|     *--100----*---98----*----*----*---94----*---92----*
     |     |                                                 |
   +4|   102    *----*----*---62----*---60----*----*----*   90
     |     |    |                                       |    |
   +3|     *    *    *---36----*---34----*---32----*    *    *
     |     |    |    |                             |    |    |
   +2|   104    *   38    *---16----*---14----*   30    *   88
     |     |    |    |    |                   |    |    |    |
   +1|     *   68    *   18    5----*----3   12    *   54    *
     |     |    |    |    |    |         |    |    |    |    |
    0|     *    *   40    *    *    *----*    *   28    *    *
     |     |    |    |    |    |              |    |    |    |
   -1|     *   70    *   20    7----*----9---10    *   52    *
     |     |    |    |    |                        |    |    |
   -2|   108    *   42    *---22----*---24----*---26    *   84
     |     |    |    |                                  |    |
   -3|     *    *    *---44----*---46----*---48----*----*    *
     |     |    |                                            |
    4|   110    *----*----*---76----*---78----*----*----*---82
     |     |
    5|     *--112----*--114----*----*----*--118----*--120----*
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, PARI program for A308412
• Eric Weisstein's World of Mathematics, Gaussian Prime

Crossrefs

Cf. A174344, A274923.

Programs

Maple

SP:= proc(n) option remember; local k;
k:=floor(sqrt(4*n-7)) mod 4;
procname(n-1) -I*exp(I*k*Pi/2)
end proc:
SP(1):= 0:
select(i -> GaussInt:-GIprime(SP(i)), [$1..1000]); # Robert Israel, May 20 2024

PARI

\\ See Links section.