A218858 - OEIS

%I #15 May 03 2013 19:01:32

%S 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,

%T 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,

%U 112,0,64,0,76,0,88,0,56,0,136,0,92,0,80,0,76,0,88,0

%N Number of Gaussian primes at taxicab distance n from the origin.

%C 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.

%C The arithmetic derivative of Gaussian primes is either 1, -1, I, or -I.

%H T. D. Noe, <a href="/A218858/b218858.txt">Table of n, a(n) for n = 0..10000</a>

%H T. D. Noe, <a href="/A218858/a218858.png">Linear plot</a>

%e 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.

%t 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}]

%Y Cf. A055025 (norms of Gaussian primes).

%Y Cf. A222593 (first-quadrant Gaussian primes).

%Y Cf. A225071, A225072 (number of terms at an odd distance from the origin).

%K nonn

%O 0,3

%A _T. D. Noe_, Nov 12 2012