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A273294 Least nonnegative integer m such that there are nonnegative integers x,y,z,w for which x^2 + y^2 + z^2 + w^2 = n and x + 3*y + 5*z = m^2.

%I A273294 #7 May 19 2016 08:53:02
%S A273294 0,0,1,2,0,1,2,3,4,0,1,2,3,3,3,4,0,1,2,3,2,3,3,4,4,0,1,2,3,3,4,4,2,3,
%T A273294 3,4,0,1,2,3,4,2,3,6,4,3,3,6,4,0,1,2,2,3,5,4,4,4,3,4,5,5,3,4,0,1,2,3,
%U A273294 4,5,4,6,4,3,4,4,4,3,4,4,2
%N A273294 Least nonnegative integer m such that there are nonnegative integers x,y,z,w for which x^2 + y^2 + z^2 + w^2 = n and x + 3*y + 5*z = m^2.
%C A273294 Clearly, a(n) = 0 if n is a square. Part (i) of the conjecture in A271518 implies that a(n) always exists.
%C A273294 For more conjectural refinements of Lagrange's four-square theorem, one may consult arXiv:1604.06723.
%H A273294 Zhi-Wei Sun, <a href="/A273294/b273294.txt">Table of n, a(n) for n = 0..10000</a>
%H A273294 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1604.06723">Refining Lagrange's four-square theorem</a>, arXiv:1604.06723 [math.GM], 2016.
%e A273294 a(1) = 0 since 1 = 0^2 + 0^2 + 0^2 + 1^2 with 0 + 3*0 + 5*0 = 0^2.
%e A273294 a(2) = 1 since 2 = 1^2 + 0^2 + 0^2 + 1^2 with 1 + 3*0 + 5*0 = 1^2.
%e A273294 a(3) = 2 since 3 = 1^2 + 1^2 + 0^2 + 1^2 with 1 + 3*1 + 5*0 = 2^2.
%e A273294 a(3812) = 11 since 3812 = 37^2 + 3^2 + 15^2 + 47^2 with 37 + 3*3 + 5*15 = 11^2.
%e A273294 a(3840) = 16 since 3840 = 48^2 + 16^2 + 32^2 + 16^2 with 48 + 3*16 + 5*32 = 16^2.
%t A273294 SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
%t A273294 Do[m=0;Label[bb];Do[If[3y+5z<=m^2&&SQ[n-y^2-z^2-(m^2-3y-5z)^2],Print[n," ",m];Goto[aa]],{y,0,Sqrt[n]},{z,0,Sqrt[n-y^2]}];m=m+1;Goto[bb];Label[aa];Continue,{n,0,80}]
%Y A273294 Cf. A000118, A000290, A260625, A261876, A262357, A267121, A268197, A268507, A269400, A270073, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824, A272084, A272332, A272351, A272620, A272888, A272977, A273021, A273107, A273108, A273110, A273134, A273278.
%K A273294 nonn
%O A273294 0,4
%A A273294 _Zhi-Wei Sun_, May 19 2016