A307531 a(n) is the greatest sum i + j + k + l where i^2 + j^2 + k^2 + l^2 = n and 0 <= i <= j <= k <= l.

0, 1, 2, 3, 4, 3, 4, 5, 4, 5, 6, 5, 6, 7, 6, 7, 8, 7, 8, 7, 8, 9, 8, 9, 8, 9, 10, 9, 10, 9, 10, 11, 8, 11, 10, 11, 12, 11, 12, 11, 12, 11, 12, 13, 12, 13, 12, 13, 12, 13, 14, 13, 14, 13, 14, 13, 12, 15, 14, 15, 14, 15, 14, 15, 16, 15, 16, 15, 16, 15, 16, 15

Offset: 0

Author

Rémy Sigrist, Apr 13 2019

Keywords

• nonn

Comments

The sequence is well defined as every nonnegative integer can be represented as a sum of four squares in at least one way.

It appears that a(n^2) = 2*n if n is even and 2*n-1 if n is odd. - Robert Israel, Apr 14 2019

Examples

			For n = 34:
- 34 can be expressed in 4 ways as a sum of four squares:
    i^2 + j^2 + k^2 + l^2   i+j+k+l
    ---------------------   -------
    0^2 + 0^2 + 3^2 + 5^2         8
    0^2 + 3^2 + 3^2 + 4^2        10
    1^2 + 1^2 + 4^2 + 4^2        10
    1^2 + 2^2 + 2^2 + 5^2        10
- a(34) = max(8, 10) = 10.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• Rémy Sigrist, C program for A307531
• Wikipedia, Lagrange's four-square theorem

Crossrefs

See A307510 for the multiplicative variant.

Cf. A002635, A354777.

Programs

C

See Links section.

Maple

g:= proc(n,k) option remember; local a;
  if k = 1 then if issqr(n) then return sqrt(n) else return -infinity fi fi;
  max(seq(a+procname(n-a^2,k-1),a=0..floor(sqrt(n))))
end proc:
seq(g(n,4), n=0..100); # Robert Israel, Apr 14 2019

Mathematica

Array[Max[Total /@ PowersRepresentations[#, 4, 2]] &, 68, 0] (* Michael De Vlieger, Apr 13 2019 *)