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A222298 Length of the Gaussian prime spiral beginning at the n-th positive real Gaussian prime (A002145).

%I A222298 #38 Mar 04 2013 23:54:35
%S A222298 12,12,260,12,236,28,28,28,28,236,20,44,44,20,20,36,76,12,12,4,12,4,
%T A222298 36,36,36,3276,76,36,36,3276,84,20,12,12,20,36,36,2444,2444,36,44,
%U A222298 1356,156,28,12,220,12,12,84,12,132,28,68,36,36,1044,20,20,28,1044,20
%N A222298 Length of the Gaussian prime spiral beginning at the n-th positive real Gaussian prime (A002145).
%C A222298 The Gaussian prime spiral is described in the short note by O'Rourke and Wagon. It is not known if every iteration is a closed loop. See A222299 for the number of distinct primes on the spiral. See A222300 for the length of the spiral (which is the same as the number of numbers tested for primality, without memory).
%C A222298 This idea can be extended to any Gaussian prime. Sequences A222594, A222595, and A222596 show the results for first-quadrant Gaussian primes. - _T. D. Noe_, Feb 27 2013
%D A222298 Joseph O'Rourke and Stan Wagon, Gaussian prime spirals, Mathematics Magazine, vol. 86, no. 1 (2013), p. 14.
%H A222298 T. D. Noe, <a href="/A222298/b222298.txt">Table of n, a(n) for n = 1..1000</a>
%H A222298 T. D. Noe, <a href="/A222298/a222298_5.png">Plot beginning with 11</a> (similar to the cover of Mathematics Magazine, vol. 86, no. 1 (2013))
%H A222298 Joseph O'Rourke, <a href="http://mathoverflow.net/questions/91423/gaussian-prime-spirals">MathOverflow: Gaussian prime spirals</a>
%e A222298 The loop beginning with 31 is {31, 43, 43 - 8i, 37 - 8i, 37 - 2i, 45 - 2i, 45 - 8i, 43 - 8i, 43, 47, 47 - 2i, 45 - 2i, 45 + 2i, 47 + 2i, 47, 43, 43 + 8i, 45 + 8i, 45 + 2i, 37 + 2i, 37 + 8i, 43 + 8i, 43, 31, 31 + 4i, 41 + 4i, 41 - 4i, 31 - 4i, 31}. The first and last numbers are the same. So only one is counted.
%t A222298 loop[n_] := Module[{p = n, direction = 1}, lst = {n}; While[While[p = p + direction; ! PrimeQ[p, GaussianIntegers -> True]]; direction = direction*(-I); AppendTo[lst, p]; ! (p == n && direction == 1)]; Length[lst]]; cp = Select[Range[1000], PrimeQ[#, GaussianIntegers -> True] &]; Table[loop[p]-1, {p, cp}]
%K A222298 nonn
%O A222298 1,1
%A A222298 _T. D. Noe_, Feb 25 2013