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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.

%I A307510 #19 Apr 15 2019 16:27:01
%S A307510 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,
%T A307510 24,54,0,48,20,36,81,40,72,30,64,0,60,108,45,96,40,90,48,80,144,60,
%U A307510 135,72,120,54,0,192,108,180,96,160,72,162,256,144,240,100
%N 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.
%C A307510 The sequence is well defined as every nonnegative integer can be represented as a sum of four squares in at least one way.
%H A307510 Robert Israel, <a href="/A307510/b307510.txt">Table of n, a(n) for n = 0..10000</a>
%H A307510 Rémy Sigrist, <a href="/A307510/a307510.png">Colored scatterplot of the first 20000 terms</a> (where the color is function of the parity of n)
%H A307510 Rémy Sigrist, <a href="/A307510/a307510.txt">C program for A307510</a>
%H A307510 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lagrange%27s_four-square_theorem">Lagrange's four-square theorem</a>
%F A307510 a(n) = 0 iff n belongs to A000534.
%F A307510 a(n) <= (n/4)^2, with equality if and only if n is an even square. - _Robert Israel_, Apr 15 2019
%e A307510 For n = 34:
%e A307510 - 34 can be expressed in 4 ways as a sum of four squares:
%e A307510     i^2 + j^2 + k^2 + l^2   i*j*k*l
%e A307510     ---------------------   -------
%e A307510     0^2 + 0^2 + 3^2 + 5^2         0
%e A307510     0^2 + 3^2 + 3^2 + 4^2         0
%e A307510     1^2 + 1^2 + 4^2 + 4^2        16
%e A307510     1^2 + 2^2 + 2^2 + 5^2        20
%e A307510 - a(34) = max(0, 16, 20) = 20.
%p A307510 g:= proc(n, k) option remember; local a;
%p A307510   if k = 1 then if issqr(n) then return sqrt(n) else return -infinity fi fi;
%p A307510   max(0,seq(a*procname(n-a^2, k-1), a=1..floor(sqrt(n))))
%p A307510 end proc:
%p A307510 seq(g(n, 4), n=0..100); # _Robert Israel_, Apr 15 2019
%t A307510 Array[Max[Times @@ # & /@ PowersRepresentations[#, 4, 2]] &, 68, 0] (* _Michael De Vlieger_, Apr 13 2019 *)
%o A307510 (C) See Links section.
%Y A307510 See A307531 for the additive variant.
%Y A307510 Cf. A000534, A002635, A122922.
%K A307510 nonn,look
%O A307510 0,8
%A A307510 _Rémy Sigrist_, Apr 11 2019