A295849 Number of nonnegative solutions to gcd(x,y,z) = 1 and x^2 + y^2 + z^2 <= n.

0, 3, 6, 7, 7, 13, 16, 16, 16, 19, 25, 28, 28, 34, 40, 40, 40, 49, 52, 55, 55, 61, 64, 64, 64, 70, 82, 85, 85, 97, 103, 103, 103, 109, 118, 124, 124, 130, 139, 139, 139, 154, 160, 163, 163, 169, 175, 175, 175, 181, 193, 199, 199, 211, 220, 220, 220, 226, 232, 241

Offset: 0

Author

Seiichi Manyama, Nov 29 2017

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000606, A048134, A295820, A295848.

Programs

Maple

N:= 100:
V:= Vector(N):
for x from 0 to floor(sqrt(N/3)) do
  for y from x to floor(sqrt((N-x^2)/2)) do
    for z from y to floor(sqrt(N-x^2-y^2)) do
      if igcd(x,y,z) = 1 then
        r:= x^2 + y^2 + z^2;
        m:= nops({x,y,z});
        if m=3 then V[r]:= V[r]+6
        elif m=2 then V[r]:= V[r]+3
        else V[r]:= V[r]+1
        fi
      fi
od od od:
0,op(ListTools:-PartialSums(convert(V,list))); # Robert Israel, Nov 30 2017

Mathematica

a[n_] := Sum[Boole[GCD[i, j, k] == 1], {i, 0, Sqrt[n]}, {j, 0, Sqrt[n - i^2]}, {k, 0, Sqrt[n - i^2 - j^2]}];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Jul 07 2018, after Andrew Howroyd *)

PARI

a(n) = {sum(i=0, sqrtint(n), sum(j=0, sqrtint(n-i^2), sum(k=0, sqrtint(n-i^2-j^2), gcd([i, j, k]) == 1)))} \\ Andrew Howroyd, Dec 12 2017

Formula

a(n) = a(n-1) + A295848(n) for n > 0.