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

Author

T. D. Noe, Mar 22 2014

Keywords

• nonn

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

Examples

			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)
		

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

Crossrefs

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

Cf. A300587, A300588.

Programs

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] &]

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

Extensions

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

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