A062711 - OEIS

%I #19 Mar 13 2023 07:19:27

%S 0,1,4,6,8,10,15,19,21,25,32,34,38,44,46,52,60,66,73,79,87,93,98,104,

%T 114,122,128,138,146,154,163,173,181,193,203,213,221,231,239,245,259,

%U 273,280,294,304,316,327,343,359,369

%N Number of prime Gaussian integers z=a+bi with |z|<=n.

%H T. D. Noe, <a href="/A062711/b062711.txt">Table of n, a(n) for n=1..1000</a>

%H <a href="/index/Ga#gaussians">Index entries for Gaussian integers and primes</a>

%F Two prime Gaussian integers are not counted separately if they are associated, i.e. if their quotient is a unit (1, i, -1 or -i).

%F Similar to the ordinary prime number theorem (see A000720) we have the asymptotic expression: a(n) ~ n^2/(2 * log(n)) - Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 16 2001

%F a(1)=0, a(n)=1+A066339(n^2)+A066490(n) for n>0. - _T. D. Noe_, Feb 20 2007

%t m = 50;

%t t = Table[x + y I, {x, -m, m}, {y, -m, m}] // Flatten[#, 1]& // Select[#, PrimeQ[#, GaussianIntegers -> True]& ]& // Sort // DeleteDuplicates[#, Abs[#1] == Abs[#2] && MatchQ[#1 /#2 , 1|-1|I|-I]& ]&;

%t a[n_] := Select[t, Abs[#] <= n&] // Length;

%t Array[a, m] (* _Jean-François Alcover_, Jul 29 2016 *)

%Y Cf. A000328, A062327, A000720.

%K nonn,nice

%O 1,3

%A _Reiner Martin_, Jul 14 2001