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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.
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%I #20 Oct 16 2024 12:04:21

%S 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,

%T 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,

%U 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

%N 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.

%H Robert Israel, <a href="/A375202/b375202.txt">Table of n, a(n) for n = 0..10000</a>

%H Mathematics StackExchange, <a href="https://math.stackexchange.com/questions/4985056/large-smallest-square-in-a-sum-of-three-squares#4985056">Large smallest square in a sum of three squares?</a>

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

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

%e 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.

%p f:= proc(n) local q,x,y,z;

%p if n/4^padic:-ordp(n,4) mod 8 = 7 then return -1 fi;

%p for x from 0 while 3*x^2 <= n do

%p if [isolve(y^2 + z^2 = n - x^2)] <> [] then return x fi

%p od;

%p end proc;

%p map(f, [$0..100]);

%o (Python)

%o from math import isqrt

%o from sympy import factorint

%o def A375202(n):

%o v = (~n & n-1).bit_length()

%o if v&1^1 and n>>v&7==7: return -1

%o for x in range(isqrt(n//3)+1):

%o if not any(e&1 and p&3==3 for p, e in factorint(n-x**2).items()):

%o return x # _Chai Wah Wu_, Oct 16 2024

%Y Cf. A064874, A375203, A375204.

%K sign

%O 0,13

%A _Robert Israel_, Oct 15 2024