A375203 -id:A375203 - cp's OEIS frontend

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

Robert Israel, Oct 15 2024

sign

			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.

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

Cf. A064874, A375203, A375204.

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]);

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

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

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

A375204 Record values in A375202.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 18, 20, 24, 32, 36, 40, 48, 64, 72, 80, 96, 128, 144, 160, 192, 256, 288, 320, 384, 512, 576, 640, 768, 1024, 1152, 1280, 1536, 2048, 2304, 2560, 3072, 4096, 4608, 5120, 6144, 8192, 9216, 10240, 12288, 16384

Offset: 1

Robert Israel, Oct 15 2024

Numbers k such that k = A375202(m) for some m such that A375202(j) < k for all j < m.

Conjectures: All powers of 2 (A000079), 3*(powers of 2) (A007283) and 5*(powers of 2) (A020714) are terms. All terms are 5-smooth (A051037). - Chai Wah Wu, Oct 16 2024

			a(3) = 2 because A375202(12) = 2 and A375202(j) <= 1 for j < 12.

Mathematics StackExchange, Large smallest square in a sum of three squares?

Cf. A000079, A007283, A020714, A051037, A375202, A375203.

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:
V:= NULL:count:= 0: m:= -1;
for i from 0 while count < 39 do
  v:= f(i);
  if v > m then
    V:= V, v; m:= v; count:=count+1
  fi
od:
V;

from itertools import count, islice
from math import isqrt
from sympy import factorint
def A375204_gen(): # generator of terms
    c = -1
    for n in count(0):
        v = (~n & n-1).bit_length()
        if v&1 or n>>v&7!=7:
            a = next(x 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()))
            if a>c:
                yield a
                c = a
A375204_list = list(islice(A375204_gen(),20)) # Chai Wah Wu, Oct 16 2024

a(n) = A375292(A375203(n)).

a(35)-a(48) from Chai Wah Wu, Oct 16 2024

a(49)-a(52) from Chai Wah Wu, Oct 17 2024

cp's OEIS Frontend

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.

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