A307510 a(n) is the greatest product i*j*k*l where i^2 + j^2 + k^2 + l^2 = n and 0 <= i <= j <= k <= l.

0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 4, 0, 3, 8, 0, 6, 16, 0, 12, 4, 9, 24, 8, 18, 0, 16, 36, 12, 32, 0, 24, 54, 0, 48, 20, 36, 81, 40, 72, 30, 64, 0, 60, 108, 45, 96, 40, 90, 48, 80, 144, 60, 135, 72, 120, 54, 0, 192, 108, 180, 96, 160, 72, 162, 256, 144, 240, 100

Offset

0,8

Comments

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

Links

Robert Israel, Table of n, a(n) for n = 0..10000

Rémy Sigrist, Colored scatterplot of the first 20000 terms (where the color is function of the parity of n)

Rémy Sigrist, C program for A307510

Wikipedia, Lagrange's four-square theorem

Formula

a(n) = 0 iff n belongs to A000534.

a(n) <= (n/4)^2, with equality if and only if n is an even square. - Robert Israel, Apr 15 2019

Example

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         0
    0^2 + 3^2 + 3^2 + 4^2         0
    1^2 + 1^2 + 4^2 + 4^2        16
    1^2 + 2^2 + 2^2 + 5^2        20
- a(34) = max(0, 16, 20) = 20.

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(0, seq(a*procname(n-a^2, k-1), a=1..floor(sqrt(n))))
end proc:
seq(g(n, 4), n=0..100); # Robert Israel, Apr 15 2019

Mathematica

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

Prog

(C) See Links section.

Crossrefs

See A307531 for the additive variant.

Cf. A000534, A002635, A122922.

Sequence in context: A084658 A326404 A358702 * A323885 A137513 A365714

Adjacent sequences: A307507 A307508 A307509 * A307511 A307512 A307513

Keyword

nonn,look

Author

Rémy Sigrist, Apr 11 2019

Status

approved