A225071 -id:A225071 - cp's OEIS frontend

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

T. D. Noe, Nov 12 2012

nonn

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.

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

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).

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}]

A225072 Number of first-quadrant Gaussian primes at taxicab distance 2n-1 from the origin.

Original entry on oeis.org

0, 3, 4, 5, 4, 7, 6, 8, 8, 9, 6, 9, 16, 8, 12, 11, 8, 18, 16, 12, 18, 15, 14, 15, 10, 14, 18, 28, 16, 19, 22, 14, 34, 23, 20, 19, 22, 18, 16, 27, 18, 31, 40, 22, 28, 26, 16, 36, 28, 20, 36, 33, 20, 35, 32, 26, 40, 40, 26, 28, 34, 24, 46, 37, 28, 45, 30, 34, 36

Offset: 1

T. D. Noe, May 03 2013

nonn

Except for 1+I, 1-I, -1+I, and -1-I, all Gaussian primes are an odd taxicab distance from the origin. Primes on the x- and y-axis are counted only once. That is, although p and p*I are Gaussian primes (for primes p in A002145), we count only p as being a first-quadrant Gaussian prime.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A002145, A218858, A225071.

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, 1, 200, 2}]/4

cp's OEIS Frontend

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

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