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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 (list; graph; refs; listen; history; text; internal format)
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, Table of n, a(n) for n = 1..1000

T. D. Noe, Plot beginning with 11 (similar to the cover of Mathematics Magazine, vol. 86, no. 1 (2013))

Joseph O'Rourke, MathOverflow: Gaussian prime spirals

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

Adjacent sequences:  A222295 A222296 A222297 * A222299 A222300 A222301

KEYWORD

nonn

AUTHOR

T. D. Noe, Feb 25 2013

STATUS

approved

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Last modified September 24 14:46 EDT 2021. Contains 347643 sequences. (Running on oeis4.)