A239621 - OEIS

%I #53 Feb 10 2026 19:52:57

%S 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,

%T 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,

%U 14,9,16,5,17,2,13,12,14,11,16,9,18,5,17,8,19,0

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

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

%C 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

%H T. D. Noe, <a href="/A239621/b239621.txt">Table of n, a(n) for n = 1..10106</a> (5053 pairs)

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GaussianPrime.html">Gaussian prime</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Complex_number">Complex Number</a>

%e From _M. F. Hasler_, Mar 09 2018: (Start)

%e Sorted by norm, the smallest Gaussian primes z = x + iy in the first half-quadrant x >= y >= 0 are:

%e   a(1) + i*a(2) = 1 + i;

%e   a(3) + i*a(4) = 2 + i;

%e   a(5) + i*a(6) = 3;

%e   ... (End)

%t 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] &]

%o (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

%o (PARI) list(maxnorm)= my(r=List(), t); for(x=1, sqrtint(maxnorm), forstep(y=(1==x) || !(x%2), x, 2, if((y && (t=x^2+y^2)<=maxnorm && isprime(t)) || (!y && (3==x%4) && isprime(x)), listput(~r, [x, y])))); vecsort(Vec(r), t->t[1]^2+t[2]^2) \\ _Ruud H.G. van Tol_, Feb 02 2026

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

%Y Cf. A300587, A300588.

%K nonn

%O 1,3

%A _T. D. Noe_, Mar 22 2014

%E Name changed and in cf. complex -> Gaussian - _Wolfdieter Lang_, Mar 25 2014

%E Name edited by _M. F. Hasler_, Mar 09 2018