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Number of coprime pairs (x,y) with x^2 + y^2 <= n.
3

%I #25 Aug 02 2023 05:21:06

%S 0,4,8,8,8,16,16,16,16,16,24,24,24,32,32,32,32,40,40,40,40,40,40,40,

%T 40,48,56,56,56,64,64,64,64,64,72,72,72,80,80,80,80,88,88,88,88,88,88,

%U 88,88,88,96,96,96,104,104,104,104,104,112,112,112,120,120,120,120

%N Number of coprime pairs (x,y) with x^2 + y^2 <= n.

%H Seiichi Manyama, <a href="/A304651/b304651.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) = a(n-1) + 4*A000089(n) for n > 0.

%e a(2) = 8 counts (x,y) = (-1,-1), (-1,0), (-1,1), (0,-1), (0,1), (1,-1), (1,0) and (1,1).

%t a89[n_] := a89[n] = Product[{p, e} = pe; Which[p < 3 && e == 1, 1, p == 2 && e > 1, 0, Mod[p, 4] == 1, 2, Mod[p, 4] == 3, 0, True, a89[p^e]], {pe, FactorInteger[n]}];

%t a[n_] := a[n] = If[n == 0, 0, a[n-1] + 4 a89[n]];

%t Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Aug 02 2023 *)

%Y Cf. A000089, A175341, A305117.

%K nonn

%O 0,2

%A _Seiichi Manyama_, May 26 2018