A275738 - OEIS

%I #13 Jun 10 2018 08:19:56

%S 1,1,1,2,3,1,1,1,3,3,1,3,4,1,1,2,3,3,2,5,5,1,1,1,5,3,3,5,3,2,2,1,2,4,

%T 2,7,7,1,2,3,5,3,2,3,8,3,1,3,4,4,3,9,6,3,3,1,4,4,1,6,5,2,3,2,5,3,3,5,

%U 8,3,1,3,7,4,4,8,4,2,2,5

%N Number of ordered ways to write n as w^2 + x^2*(1+y^2+z^2), where w,x,y,z are nonnegative integers with x > 0, y <= z and y == z (mod 2).

%C Conjecture: For any n > 0, we have a(n) > 0, i.e., n can be written as w^2 + x^2*(1+(z-y)^2+(y+z)^2) = w^2 + x^2*(1+2*y^2+2*z^2), where w,x,y,z are nonnegative integers with x > 0 and y <= z. Moreover, any positive integer n not equal to 449 can be written as 4^k*(1+x^2+y^2) + z^2, where k,x,y,z are nonnegative integers with x == y (mod 2).

%C This is stronger than Lagrange's four-square theorem, and we have verified it for n up to 10^6.

%C See also A275648, A275656, A275675, A275676 and A275678 for similar conjectures.

%H Zhi-Wei Sun, <a href="/A275738/b275738.txt">Table of n, a(n) for n = 1..10000</a>

%H Zhi-Wei Sun, <a href="https://arxiv.org/abs/1604.06723">Refining Lagrange's four-square theorem</a>, arXiv:1604.06723 [math.NT], 2016-2017.

%H Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2016.11.008">Refining Lagrange's four-square theorem</a>, J. Number Theory 175(2017), 167-190, 2017.

%e a(2) = 1 since 2 = 1^2 + 1^2*(1+0^2+0^2) with 0 + 0 even.

%e a(7) = 1 since 7 = 2^2 + 1^2*(1+1^2+1^2) with 1 + 1 even.

%e a(59) = 1 since 59 = 0^2 + 1^2*(1+3^2+7^2) with 3 + 7 even.

%e a(71) = 1 since 71 = 6^2 + 1^2*(1+3^2+5^2) with 3 + 5 even.

%e a(113) = 2 since 113 = 7^2 + 8^2*(1+0^2+0^2) = 8^2 + 7^2*(1+0^2+0^2) with 0 + 0 even.

%e a(143) = 1 since 143 = 6^2 + 1^2*(1+5^2+9^2) with 5 + 9 even.

%e a(191) = 1 since 191 = 10^2 + 1^2*(1+3^2+9^2) with 3 + 9 even.

%e a(449) = 3 since 449 = 18^2 + 5^2*(1+0^2+2^2) with 0 + 2 even, and 449 = 7^2 + 20^2*(1+0^2+0^2) = 20^2 + 7^2*(1+0^2+0^2) with 0 + 0 even.

%e a(497) = 1 since 497 = 15^2 + 4^2*(1+0^2+4^2) with 0 + 4 even.

%e a(2033) = 1 since 2033 = 33^2 + 4^2*(1+3^2+7^2) with 3 + 7 even.

%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]

%t Do[r=0;Do[If[SQ[n-x^2*(1+2y^2+2z^2)],r=r+1],{x,1,Sqrt[n]},{y,0,Sqrt[(n/x^2-1)/4]},{z,y,Sqrt[(n/x^2-1-2y^2)/2]}];Print[n," ",r];Continue,{n,1,80}]

%Y Cf. A000118, A000290, A271518, A275648, A275656, A275675, A275676, A275678.

%K nonn

%O 1,4

%A _Zhi-Wei Sun_, Aug 07 2016