A062711 Number of prime Gaussian integers z=a+bi with |z|<=n.

0, 1, 4, 6, 8, 10, 15, 19, 21, 25, 32, 34, 38, 44, 46, 52, 60, 66, 73, 79, 87, 93, 98, 104, 114, 122, 128, 138, 146, 154, 163, 173, 181, 193, 203, 213, 221, 231, 239, 245, 259, 273, 280, 294, 304, 316, 327, 343, 359, 369

Offset: 1

Author

Reiner Martin, Jul 14 2001

Keywords

• nonn
• nice

Links

• T. D. Noe, Table of n, a(n) for n=1..1000
• Index entries for Gaussian integers and primes

Crossrefs

Cf. A000328, A062327, A000720.

Programs

Mathematica

m = 50;
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]& ]&;
a[n_] := Select[t, Abs[#] <= n&] // Length;
Array[a, m] (* Jean-François Alcover, Jul 29 2016 *)

Formula

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).

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

a(1)=0, a(n)=1+A066339(n^2)+A066490(n) for n>0. - T. D. Noe, Feb 20 2007