A218858 Number of Gaussian primes at taxicab distance n from the origin.

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

Offset: 0

Author

T. D. Noe, Nov 12 2012

Keywords

• nonn

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.

Examples

			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.
		

Links

• T. D. Noe, Table of n, a(n) for n = 0..10000
• T. D. Noe, Linear plot

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).

Programs

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}]