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A222298
Length of the Gaussian prime spiral beginning at the n-th positive real Gaussian prime (A002145).
6
12, 12, 260, 12, 236, 28, 28, 28, 28, 236, 20, 44, 44, 20, 20, 36, 76, 12, 12, 4, 12, 4, 36, 36, 36, 3276, 76, 36, 36, 3276, 84, 20, 12, 12, 20, 36, 36, 2444, 2444, 36, 44, 1356, 156, 28, 12, 220, 12, 12, 84, 12, 132, 28, 68, 36, 36, 1044, 20, 20, 28, 1044, 20
OFFSET
1,1
COMMENTS
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).
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
REFERENCES
Joseph O'Rourke and Stan Wagon, Gaussian prime spirals, Mathematics Magazine, vol. 86, no. 1 (2013), p. 14.
LINKS
T. D. Noe, Plot beginning with 11 (similar to the cover of Mathematics Magazine, vol. 86, no. 1 (2013))
EXAMPLE
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.
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]]; cp = Select[Range[1000], PrimeQ[#, GaussianIntegers -> True] &]; Table[loop[p]-1, {p, cp}]
CROSSREFS
Sequence in context: A165833 A038338 A221796 * A122253 A156456 A077180
KEYWORD
nonn
AUTHOR
T. D. Noe, Feb 25 2013
STATUS
approved