A064874 -id:A064874 - cp's OEIS frontend

A064873 First of four sequences representing the lexicographical minimal decomposition of n in 4 squares: n = a(n)^2 + A064874(n)^2 + A064875(n)^2 + A064876(n)^2.

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0

Offset: 0

Reinhard Zumkeller, Oct 10 2001

nonn

A072401(n) = A057427(a(n)).

For k<112: a(n)=A072401(n), but A072401(112) = 1<>a(112)=2, as also A072401(112 - 1) = 1.

			a(25) = 0: 25 = a(25)^2 + A064874(25)^2 + A064875(25)^2 + A064876(25)^2 = 0 + 0 + 0 + 25 and the other decompositions (0, 0, 3, 4) and (1, 2, 2, 4) are greater than (0, 0, 0, 5).

Eric Weisstein's World of Mathematics, Square Numbers.
Index entries for sequences related to sums of squares.

Cf. A064874, A064875, A064876, A071374.

A064876 Last of four sequences representing the lexicographical minimal decomposition of n in 4 squares: n = A064873(n)^2 + A064874(n)^2 + A064875(n)^2 + a(n)^2.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 2, 3, 3, 3, 4, 4, 3, 3, 4, 4, 3, 3, 4, 5, 5, 5, 5, 5, 5, 5, 4, 4, 5, 5, 6, 6, 6, 6, 6, 5, 5, 5, 6, 6, 6, 6, 4, 7, 7, 7, 6, 7, 7, 7, 6, 7, 7, 7, 7, 6, 6, 6, 8, 8, 8, 7, 8, 8, 6, 6, 6, 8, 7, 7, 6, 8, 7, 7, 8, 9, 9, 9, 8, 9, 9, 9, 6, 8, 9, 9, 9, 8, 9, 9, 8, 9, 7, 7, 10, 10, 10

Offset: 0

Reinhard Zumkeller, Oct 10 2001

nonn

			a(18) = 3: 18 = A064873(18)^2 + A064874(18)^2 + A064875(18)^2 + a(18)^2 = 0 + 0 + 9 + 9 and the other decompositions (0, 1, 1, 4) and (1, 2, 2, 3) are greater than (0, 0, 3, 3).

Cf. A064873, A064874, A064875, A064877.

A064875 Third of four sequences representing the lexicographical minimal decomposition of n in four squares: n = A064873(n)^2 + A064874(n)^2 + a(n)^2 + A064876(n)^2.

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 1, 1, 2, 0, 1, 1, 2, 2, 2, 2, 0, 1, 3, 3, 2, 2, 3, 3, 2, 0, 1, 1, 1, 2, 2, 2, 4, 4, 3, 3, 0, 1, 1, 1, 2, 4, 4, 3, 2, 3, 3, 3, 4, 0, 1, 1, 4, 2, 2, 2, 4, 2, 3, 3, 3, 5, 5, 5, 0, 1, 1, 3, 2, 2, 5, 5, 6, 3, 5, 5, 6, 3, 5, 5, 4, 0, 1, 1, 4, 2, 2, 2, 6, 5, 3, 3, 3, 5, 3, 3, 4, 4, 7, 7, 0, 1, 1, 1, 2

Offset: 0

Reinhard Zumkeller, Oct 10 2001

nonn

			a(19) = 3: 19 = A064873(19)^2 + A064874(19)^2 + a(19)^2 + A064876(19)^2 = 0 + 1 + 9 + 9 and the other decomposition (1, 1, 1, 4) is greater than (0, 1, 3, 3).

Cf. A064873, A064874, A064876, A064877.

A064877 a(n) = n - (A064873(n) + A064874(n) + A064875(n) + A064876(n)).

Original entry on oeis.org

0, 0, 0, 0, 2, 2, 2, 2, 4, 6, 6, 6, 6, 8, 8, 8, 12, 12, 12, 12, 14, 14, 14, 14, 16, 20, 20, 20, 20, 22, 22, 22, 24, 24, 26, 26, 30, 30, 30, 30, 32, 32, 32, 32, 34, 36, 36, 36, 36, 42, 42, 42, 42, 44, 44, 44, 44, 46, 48, 48, 48, 50, 50, 50, 56, 56, 56, 54, 58, 58, 56, 56, 60, 62

Offset: 0

Reinhard Zumkeller, Oct 10 2001

nonn

Cf. A064873, A064874, A064875, A064876.

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.

Original entry on oeis.org

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).

cp's OEIS Frontend

A064873 First of four sequences representing the lexicographical minimal decomposition of n in 4 squares: n = a(n)^2 + A064874(n)^2 + A064875(n)^2 + A064876(n)^2.

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A064876 Last of four sequences representing the lexicographical minimal decomposition of n in 4 squares: n = A064873(n)^2 + A064874(n)^2 + A064875(n)^2 + a(n)^2.

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A064875 Third of four sequences representing the lexicographical minimal decomposition of n in four squares: n = A064873(n)^2 + A064874(n)^2 + a(n)^2 + A064876(n)^2.

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A064877 a(n) = n - (A064873(n) + A064874(n) + A064875(n) + A064876(n)).

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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.

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