A375202 a(n) is the least integer x >= 0 such that n = x^2 + y^2 + z^2 for some integers y, z, or -1 if there is no such x.

0, 0, 0, 1, 0, 0, 1, -1, 0, 0, 0, 1, 2, 0, 1, -1, 0, 0, 0, 1, 0, 1, 2, -1, 2, 0, 0, 1, -1, 0, 1, -1, 0, 1, 0, 1, 0, 0, 1, -1, 0, 0, 1, 3, 2, 0, 1, -1, 4, 0, 0, 1, 0, 0, 1, -1, 2, 2, 0, 1, -1, 0, 1, -1, 0, 0, 1, 3, 0, 1, 3, -1, 0, 0, 0, 1, 2, 2, 2, -1, 0, 0, 0, 1, 2, 0, 1, -1, 4, 0, 0, 1, -1, 2, 2

Offset: 0

Author

Robert Israel, Oct 15 2024

Keywords

• sign

Examples

			a(12) = 2 because 12 = 2^2 + 2^2 + 2^2 but there are no integer solutions to 12 = 0^2 + y^2 + z^2 or 12 = 1^2 + y^2 + z^2.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• Mathematics StackExchange, Large smallest square in a sum of three squares?

Crossrefs

Cf. A064874, A375203, A375204.

Programs

Maple

f:= proc(n) local q,x,y,z;
  if n/4^padic:-ordp(n,4) mod 8 = 7 then return -1 fi;
  for x from 0 while 3*x^2 <= n do
    if [isolve(y^2 + z^2 = n - x^2)] <> [] then return x fi
  od;
end proc;
map(f, [$0..100]);

Python

from math import isqrt
from sympy import factorint
def A375202(n):
    v = (~n & n-1).bit_length()
    if v&1^1 and n>>v&7==7: return -1
    for x in range(isqrt(n//3)+1):
        if not any(e&1 and p&3==3 for p, e in factorint(n-x**2).items()):
            return x # Chai Wah Wu, Oct 16 2024

Formula

a(n) = A064874(n) if a(n) >= 0.

If a(n) = -1 then a(4*n) = -1, otherwise a(4*n) = 2*a(n).