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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.

%I A375202 #20 Oct 16 2024 12:04:21
%S A375202 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 A375202 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 A375202 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 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.
%H A375202 Robert Israel, <a href="/A375202/b375202.txt">Table of n, a(n) for n = 0..10000</a>
%H A375202 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 A375202 a(n) = A064874(n) if a(n) >= 0.
%F A375202 If a(n) = -1 then a(4*n) = -1, otherwise a(4*n) = 2*a(n).
%e A375202 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 A375202 f:= proc(n) local q,x,y,z;
%p A375202   if n/4^padic:-ordp(n,4) mod 8 = 7 then return -1 fi;
%p A375202   for x from 0 while 3*x^2 <= n do
%p A375202     if [isolve(y^2 + z^2 = n - x^2)] <> [] then return x fi
%p A375202   od;
%p A375202 end proc;
%p A375202 map(f, [$0..100]);
%o A375202 (Python)
%o A375202 from math import isqrt
%o A375202 from sympy import factorint
%o A375202 def A375202(n):
%o A375202     v = (~n & n-1).bit_length()
%o A375202     if v&1^1 and n>>v&7==7: return -1
%o A375202     for x in range(isqrt(n//3)+1):
%o A375202         if not any(e&1 and p&3==3 for p, e in factorint(n-x**2).items()):
%o A375202             return x # _Chai Wah Wu_, Oct 16 2024
%Y A375202 Cf. A064874, A375203, A375204.
%K A375202 sign
%O A375202 0,13
%A A375202 _Robert Israel_, Oct 15 2024