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A295996 One quarter of number of Gaussian primes whose norm is 4*n+1 or less.

%I A295996 #30 Feb 16 2025 08:33:52
%S A295996 0,3,4,6,8,8,8,10,10,12,14,14,15,17,17,19,19,19,21,21,21,21,23,23,25,
%T A295996 27,27,29,31,31,32,32,32,32,34,34,34,36,36,38,38,38,38,40,40,42,42,42,
%U A295996 44,46,46,46,46,46,46,46,46,48,50,50,52,52,52,52,54,54,54,56
%N A295996 One quarter of number of Gaussian primes whose norm is 4*n+1 or less.
%H A295996 Seiichi Manyama, <a href="/A295996/b295996.txt">Table of n, a(n) for n = 0..10000</a>
%H A295996 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GaussianPrime.html">Gaussian Prime</a>
%e A295996 The Gaussian primes whose norm is 9 or less;
%e A295996       *        3i,
%e A295996     *   *      -1+2i, 1+2i
%e A295996   * *   * *    -2+i, -1+i, 1+i, 2+i
%e A295996 *           *  -3, 3
%e A295996   * *   * *    -2-i, -1-i, 1-i, 2-i
%e A295996     *   *      -1-2i, 1-2i
%e A295996       *        -3i
%e A295996                a(2) = 16/4 = 4.
%o A295996 (Ruby)
%o A295996 require 'prime'
%o A295996 def A(k, n)
%o A295996   ary = []
%o A295996   cnt = 0
%o A295996   k.step(4 * n + k, 4){|i|
%o A295996     cnt += 1 if i.prime?
%o A295996     ary << cnt
%o A295996   }
%o A295996   ary
%o A295996 end
%o A295996 def A295996(n)
%o A295996   ary1 = A(1, n)
%o A295996   ary3 = A(3, Math.sqrt(n).to_i) + [0]
%o A295996   [0] + (1..n).map{|i| 1 + 2 * ary1[i] + ary3[(Math.sqrt(4 * i + 1).to_i - 3) / 4]}
%o A295996 end
%o A295996 p A295996(100)
%Y A295996 Cf. A016813, A055029, A091100, A091134, A135462, A296020, A296021.
%K A295996 nonn
%O A295996 0,2
%A A295996 _Seiichi Manyama_, Dec 02 2017