A239621 Gaussian primes x + i*y, with x = a(2n-1) >= y = a(2n) >= 0, sorted by norm.

1, 1, 2, 1, 3, 0, 3, 2, 4, 1, 5, 2, 6, 1, 5, 4, 7, 0, 7, 2, 6, 5, 8, 3, 8, 5, 9, 4, 10, 1, 10, 3, 8, 7, 11, 0, 11, 4, 10, 7, 11, 6, 13, 2, 10, 9, 12, 7, 14, 1, 15, 2, 13, 8, 15, 4, 16, 1, 13, 10, 14, 9, 16, 5, 17, 2, 13, 12, 14, 11, 16, 9, 18, 5, 17, 8, 19, 0

Offset

1,3

Comments

The condition a >= b >= 0 implies that there is only one Gaussian prime for each norm. - T. D. Noe, Mar 26 2014

The real parts and imaginary parts are listed as a(2n-1) = A300587(n) and a(2n) = A300588(n), respectively. Sequence A239397 lists the pair (y, x) after each pair (x, y), except for (1, 1). - M. F. Hasler, Mar 10 2018

Links

T. D. Noe, Table of n, a(n) for n = 1..10106 (5053 pairs)

Eric Weisstein's World of Mathematics, Gaussian prime

Wikipedia, Complex Number

Example

From M. F. Hasler, Mar 09 2018: (Start)
Sorted by norm, the smallest Gaussian primes z = x + iy in the first half-quadrant x >= y >= 0 are:
a(1) + i*a(2) = 1 + i;
a(3) + i*a(4) = 2 + i;
a(5) + i*a(6) = 3;
... (End)

Mathematica

mx = 20; lst = Flatten[Table[{a, b}, {a, 0, mx}, {b, 0, a}], 1]; qq = Select[lst, Norm[#] <= mx && PrimeQ[#[[1]] + I*#[[2]], GaussianIntegers -> True] &]; Sort[qq, Norm[#1] < Norm[#2] &]

Prog

(PARI) {for(n=2, 400, f=factor(n*I)/*factor in Z[i]*/; matsize(f)[1]<=2 && vecsum(f[, 2])==2+(f[1, 1]==I) /*either I*p^2 or w*conj(w/I), maybe (1+I)^2 */ && printf("%d, ", vecsort([real(f=f[3-f[1, 2], 1]), imag(f)], , 4)))} \\ For illustrative use. - M. F. Hasler, Mar 09 2018

Crossrefs

Cf. A055025 (norms of Gaussian primes), A239397.

Cf. A300587, A300588.

Sequence in context: A122170 A066029 A141198 * A349444 A231204 A180987

Adjacent sequences: A239618 A239619 A239620 * A239622 A239623 A239624

Keyword

nonn

Author

T. D. Noe, Mar 22 2014

Extensions

Name changed and in cf. complex -> Gaussian - Wolfdieter Lang, Mar 25 2014

Name edited by M. F. Hasler, Mar 09 2018

Status

approved