A176562 -id:A176562 - cp's OEIS frontend

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

R. J. Mathar, Apr 16 2010

nonn

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

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.

Cf. A000328, A176562, A304651.

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 *)

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

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

A305117 a(n) = A304651(n)/4.

Original entry on oeis.org

0, 1, 2, 2, 2, 4, 4, 4, 4, 4, 6, 6, 6, 8, 8, 8, 8, 10, 10, 10, 10, 10, 10, 10, 10, 12, 14, 14, 14, 16, 16, 16, 16, 16, 18, 18, 18, 20, 20, 20, 20, 22, 22, 22, 22, 22, 22, 22, 22, 22, 24, 24, 24, 26, 26, 26, 26, 26, 28, 28, 28, 30, 30, 30, 30, 34, 34, 34, 34, 34, 34

Offset: 0

Seiichi Manyama, May 26 2018

nonn

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

Cf. A000089, A176562, A304651.

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

a(n) = a(n-1) + A000089(n) for n > 0.

cp's OEIS Frontend

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

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