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A351206 Least positive integer m such that n = x^4 + (y^4 + z^4 + 7*w^2)/m^4 for some nonnegative integers x,y,z,w with y <= z.

%I A351206 #11 Feb 05 2022 23:39:15
%S A351206 1,1,1,1,2,2,2,1,1,1,1,2,2,2,4,3,1,1,1,6,2,2,10,1,1,1,2,2,1,1,1,1,1,1,
%T A351206 3,4,2,4,10,1,1,2,4,2,1,1,1,3,1,3,2,2,2,2,2,1,3,4,2,2,1,1,3,1,1,1,1,2,
%U A351206 4,2,2,3,8,10,2,2,1,10,5,1,1,1,1,1,4,2,2,8,1,1,1,2,2,2,2,1,1,1,1,4,4
%N A351206 Least positive integer m such that n = x^4 + (y^4 + z^4 + 7*w^2)/m^4 for some nonnegative integers x,y,z,w with y <= z.
%C A351206 Conjecture: a(n) exists for any nonnegative integer n.
%C A351206 This implies that each nonnegative rational number can be written as 7*w^2 + x^4 + y^4 + z^4 with w,x,y,z rational numbers.
%H A351206 Zhi-Wei Sun, <a href="/A351206/b351206.txt">Table of n, a(n) for n = 0..10000</a>
%H A351206 Zhi-Wei Sun, <a href="http://maths.nju.edu.cn/~zwsun/179b.pdf">New conjectures on representations of integers (I)</a>, Nanjing Univ. J. Math. Biquarterly 34 (2017), no.2, 97-120.
%H A351206 Zhi-Wei Sun, <a href="http://arxiv.org/abs/2010.05775">Sums of four rational squares with certain restrictions</a>, arXiv:2010.05775 [math.NT], 2020-2022.
%e A351206 a(6) = 2 with 6 = 1^4 + (1^4 + 2^4 + 7*3^2)/2^4.
%e A351206 a(19) = 6 with 19 = 0^4 + (1^4 + 4^4 + 7*59^2)/6^4.
%e A351206 a(22) = 10 with 22 = 2^4 + (2^4 + 13^4 + 7*67^2)/10^4.
%e A351206 a(5797) = 20 with 5797 = 0^4 + (81^4 + 164^4 + 7*4797^2)/20^4.
%t A351206 SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t A351206 tab={};Do[m=1; Label[bb]; k=m^4; Do[If[SQ[(k*(n-x^4)-y^4-z^4)/7], tab=Append[tab,m]; Goto[aa]],  {x, 0, n^(1/4)}, {y, 0, (k*(n-x^4)/2)^(1/4)},{z,y,(k*(n-x^4)-y^4)^(1/4)}]; m=m+1; Goto[bb]; Label[aa], {n,0,100}];Print[tab]
%Y A351206 Cf. A000290, A000583, A214891, A348890, A346643, A347827, A347865, A349942, A349943, A350714, A350857, A350860.
%K A351206 nonn
%O A351206 0,5
%A A351206 _Zhi-Wei Sun_, Feb 04 2022