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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.
3
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
0,13
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).
EXAMPLE
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.
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]);
PROG
(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
CROSSREFS
KEYWORD
sign
AUTHOR
Robert Israel, Oct 15 2024
STATUS
approved