A218859 -id:A218859 - 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}]

A222593 Pairs of numbers (a,b) such that a + b*I is a first-quadrant Gaussian prime, ordered by a+b and then a.

Original entry on oeis.org

1, 1, 0, 3, 1, 2, 2, 1, 3, 0, 1, 4, 2, 3, 3, 2, 4, 1, 0, 7, 1, 6, 2, 5, 5, 2, 6, 1, 7, 0, 2, 7, 4, 5, 5, 4, 7, 2, 0, 11, 1, 10, 3, 8, 5, 6, 6, 5, 8, 3, 10, 1, 11, 0, 3, 10, 4, 9, 5, 8, 8, 5, 9, 4, 10, 3, 1, 14, 2, 13, 4, 11, 7, 8, 8, 7, 11, 4, 13, 2, 14, 1

Offset: 1

T. D. Noe, Feb 27 2013

nonn

Gaussian primes on the positive real and imaginary axes are included. Note that the primes are ordered by their taxicab distance from the origin.

T. D. Noe, Table of n, a(n) for n = 1..1662 (taxicab distance less than 100)

Cf. A218858, A218859 (number of Gaussian primes at taxicab distance n).

nn = 20; t = {}; Do[If[PrimeQ[i + (j - i) I, GaussianIntegers -> True], AppendTo[t, {i, j-i}]], {j, 0, nn}, {i, 0, j}]; Flatten[t]

cp's OEIS Frontend

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

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