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Length of the Gaussian prime spiral beginning at the n-th first-quadrant Gaussian prime ( A222593).
+20
3
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
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.
REFERENCES
Joseph O'Rourke and Stan Wagon, Gaussian prime spirals, Mathematics Magazine, vol. 86, no. 1 (2013), p. 14.
EXAMPLE
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.
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]}]
CROSSREFS
Cf. A222298 (spiral lengths beginning at the n-th positive real Gaussian prime).
Number of different Gaussian primes in the Gaussian prime spiral beginning at the n-th first-quadrant Gaussian prime ( A222593).
+20
3
4, 24, 24, 4, 8, 22, 22, 8, 4, 8, 4, 22, 8, 4, 10, 92, 4, 92, 10, 10, 22, 22, 10, 22, 22, 4, 172, 10, 10, 92, 10, 10, 92, 92, 4, 10, 4, 10, 172, 4, 4, 10, 172, 92, 10, 172, 172, 4, 4, 172, 172, 172, 92, 10, 92, 28, 172, 4, 12, 92, 10, 10, 172, 92, 4, 12, 172, 28
COMMENTS
This is the idea of A222299 extended to first-quadrant Gaussian primes. The first odd number is a(79) = 29.
REFERENCES
Joseph O'Rourke and Stan Wagon, Gaussian prime spirals, Mathematics Magazine, vol. 86, no. 1 (2013), p. 14.
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]]]; Length[Union[lst]], {n, Length[ps]}]
CROSSREFS
Cf. A222298 (spiral lengths beginning at the n-th positive real Gaussian prime).
Number of Gaussian primes at taxicab distance n from the origin.
+10
5
0, 0, 4, 12, 0, 16, 0, 20, 0, 16, 0, 28, 0, 24, 0, 32, 0, 32, 0, 36, 0, 24, 0, 36, 0, 64, 0, 32, 0, 48, 0, 44, 0, 32, 0, 72, 0, 64, 0, 48, 0, 72, 0, 60, 0, 56, 0, 60, 0, 40, 0, 56, 0, 72, 0, 112, 0, 64, 0, 76, 0, 88, 0, 56, 0, 136, 0, 92, 0, 80, 0, 76, 0, 88, 0
COMMENTS
Except for n = 2, there are no Gaussian primes at an even taxicab distance from the origin. All terms are multiples of 4. See A218859 for this sequence divided by 4. The arithmetic derivative of Gaussian primes is either 1, -1, I, or -I.
EXAMPLE
In the taxicab distance, the four Gaussian primes closest to the origin are 1+I, -1+I, -i-I, and 1-I. The 12 at taxicab distance 3 are the four reflections of 3, 2+I, and 1+2I.
MATHEMATICA
Table[cnt = 0; Do[If[PrimeQ[n - i + I*i, GaussianIntegers -> True], cnt++], {i, 0, n}]; Do[If[PrimeQ[i - n + I*i, GaussianIntegers -> True], cnt++], {i, n - 1, 0, -1}]; Do[If[PrimeQ[i - n - I*i, GaussianIntegers -> True], cnt++], {i, 1, n}]; Do[If[PrimeQ[n - i - I*i, GaussianIntegers -> True], cnt++], {i, n - 1, 1, -1}]; cnt, {n, 0, 100}]
CROSSREFS
Cf. A055025 (norms of Gaussian primes). Cf. A222593 (first-quadrant Gaussian primes). Cf. A225071, A225072 (number of terms at an odd distance from the origin).
0, 0, 1, 3, 0, 4, 0, 5, 0, 4, 0, 7, 0, 6, 0, 8, 0, 8, 0, 9, 0, 6, 0, 9, 0, 16, 0, 8, 0, 12, 0, 11, 0, 8, 0, 18, 0, 16, 0, 12, 0, 18, 0, 15, 0, 14, 0, 15, 0, 10, 0, 14, 0, 18, 0, 28, 0, 16, 0, 19, 0, 22, 0, 14, 0, 34, 0, 23, 0, 20, 0, 19, 0, 22, 0, 18, 0, 16, 0
COMMENTS
Essentially the number of first-quadrant Gaussian primes at taxicab distance n.
CROSSREFS
Cf. A222593 (first-quadrant Gaussian primes).
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