A175341 Number of coprime pairs (x,y) with x^2+y^2 <= n^2.

0, 4, 8, 16, 32, 48, 72, 88, 120, 152, 192, 224, 264, 312, 384, 440, 480, 544, 616, 672, 768, 832, 928, 1000, 1112, 1192, 1280, 1384, 1488, 1584, 1704, 1816, 1960, 2072, 2224, 2344, 2480, 2600, 2752, 2912, 3064, 3184, 3360, 3480, 3696, 3856, 4016, 4176

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Wikipedia, Gauss circle problem

J. Wu, On the primitive circle problem, Monatsh. Math. 135 (2002), 69.

W. G. Zhai and X.D. Cao, On the number of coprime integer pairs within a circle, Acta Arith. 90 (1999), 1.

Formula

a(n) = 4*A176562(n). - R. J. Mathar, May 07 2010

a(n) = A304651(n^2). - Seiichi Manyama, May 26 2018

Example

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

Mathematica

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]}];
b[n_] := b[n] = If[n == 0, 0, b[n-1] + 4 a89[n]];
a[n_] := b[n^2];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Aug 02 2023 *)

Crossrefs

Cf. A000328, A176562, A304651.

Sequence in context: A101434 A293780 A048168 * A317705 A318692 A291441

Adjacent sequences: A175338 A175339 A175340 * A175342 A175343 A175344

Keyword

nonn

Author

R. J. Mathar, Apr 16 2010

Status

approved