A222594 Length of the Gaussian prime spiral beginning at the n-th first-quadrant Gaussian prime (A222593).

4, 28, 28, 4, 12, 28, 28, 12, 4, 12, 4, 28, 12, 4, 12, 100, 4, 100, 12, 12, 28, 28, 12, 28, 28, 4, 260, 12, 12, 100, 12, 12, 100, 100, 4, 12, 4, 12, 260, 4, 4, 12, 260, 100, 12, 260, 260, 4, 4, 260, 260, 260, 100, 12, 100, 28, 260, 4, 12, 100, 12, 12, 260

Offset: 1

Author

T. D. Noe, Feb 27 2013

Keywords

• nonn

Comments

This is the idea of A222298 extended to first-quadrant Gaussian primes (A222593). It appears that all multiples of 4 eventually appear as a length.

Examples

			The smallest such prime is 1 + i. The spiral is {1 + i, 2 + i, 2 - i, 1 - i, 1 + i}, which consists of only Gaussian primes.
		

References

• Joseph O'Rourke and Stan Wagon, Gaussian prime spirals, Mathematics Magazine, vol. 86, no. 1 (2013), p. 14.

Links

• T. D. Noe, Table of n, a(n) for n = 1..2829

Crossrefs

Cf. A222298 (spiral lengths beginning at the n-th positive real Gaussian prime).

Programs

Mathematica

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]]; nn = 20; ps = {}; Do[If[PrimeQ[i + (j - i) I, GaussianIntegers -> True], AppendTo[ps, i + (j-i)*I]], {j, 0, nn}, {i, 0, j}]; Table[loop[ps[[n]]] - 1, {n, Length[ps]}]