A175341 - OEIS

%I #23 Aug 02 2023 07:19:37

%S 0,4,8,16,32,48,72,88,120,152,192,224,264,312,384,440,480,544,616,672,

%T 768,832,928,1000,1112,1192,1280,1384,1488,1584,1704,1816,1960,2072,

%U 2224,2344,2480,2600,2752,2912,3064,3184,3360,3480,3696,3856,4016,4176

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

%H Seiichi Manyama, <a href="/A175341/b175341.txt">Table of n, a(n) for n = 0..10000</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Gauss&#39;s_circle_problem">Gauss circle problem</a>

%H J. Wu, <a href="http://dx.doi.org/10.1007/s006050200006">On the primitive circle problem</a>, Monatsh. Math. 135 (2002), 69.

%H W. G. Zhai and X.D. Cao, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa90/aa9011.pdf">On the number of coprime integer pairs within a circle</a>, Acta Arith. 90 (1999), 1.

%F a(n) = 4*A176562(n). - _R. J. Mathar_, May 07 2010

%F a(n) = A304651(n^2). - _Seiichi Manyama_, May 26 2018

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

%t a[n_] := b[n^2];

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

%Y Cf. A000328, A176562, A304651.

%K nonn

%O 0,2

%A _R. J. Mathar_, Apr 16 2010