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%I #89 May 22 2024 10:20:33
%S 3,5,7,9,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,
%T 52,54,60,62,68,70,76,78,82,84,88,90,92,94,98,100,102,104,108,110,112,
%U 114,118,120,122,126,128,132,134,138,140,144,146,150,152,156,158
%N Indices of Gaussian primes on a square spiral.
%C These are the numbers k > 0 such that A174344(k) + i*A274923(k) is a Gaussian prime (where i denotes the imaginary unit).
%C 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.
%C All terms except the first four are even.
%H Robert Israel, <a href="/A308412/b308412.txt">Table of n, a(n) for n = 1..10000</a>
%H Rémy Sigrist, <a href="/A308412/a308412_1.gp.txt">PARI program for A308412</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GaussianPrime.html">Gaussian Prime</a>
%e The first terms displayed on the center of a counterclockwise square spiral are:
%e y\x| -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5
%e ---+--------------------------------------------------------
%e +5| *--100----*---98----*----*----*---94----*---92----*
%e | | |
%e +4| 102 *----*----*---62----*---60----*----*----* 90
%e | | | | |
%e +3| * * *---36----*---34----*---32----* * *
%e | | | | | | |
%e +2| 104 * 38 *---16----*---14----* 30 * 88
%e | | | | | | | | |
%e +1| * 68 * 18 5----*----3 12 * 54 *
%e | | | | | | | | | | |
%e 0| * * 40 * * *----* * 28 * *
%e | | | | | | | | | |
%e -1| * 70 * 20 7----*----9---10 * 52 *
%e | | | | | | | |
%e -2| 108 * 42 *---22----*---24----*---26 * 84
%e | | | | | |
%e -3| * * *---44----*---46----*---48----*----* *
%e | | | |
%e 4| 110 *----*----*---76----*---78----*----*----*---82
%e | |
%e 5| *--112----*--114----*----*----*--118----*--120----*
%p SP:= proc(n) option remember; local k;
%p k:=floor(sqrt(4*n-7)) mod 4;
%p procname(n-1) -I*exp(I*k*Pi/2)
%p end proc:
%p SP(1):= 0:
%p select(i -> GaussInt:-GIprime(SP(i)), [$1..1000]); # _Robert Israel_, May 20 2024
%o (PARI) \\ See Links section.
%Y Cf. A174344, A274923.
%K nonn
%O 1,1
%A _Rémy Sigrist_, Jun 01 2019
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