A275648 - OEIS

%I #9 Aug 04 2016 06:39:28

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

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

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

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

%C Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 3, 6, 7, 11, 14, 15, 23, 38, 47, 71, 77, 143, 152, 191, 608, 2^(2k+1) (k = 0,1,2,...).

%C This is stronger than Lagrange's four-square theorem.

%C See A275656 for a stronger conjecture.

%H Zhi-Wei Sun, <a href="/A275648/b275648.txt">Table of n, a(n) for n = 1..10000</a>

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

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

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

%e a(6) = 1 since 6 = 1^2*(1+0^2+1^2) + 2^2 with 0 < 1 < 2.

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

%e a(11) = 1 since 11 = 1^2*(1+0^2+1^2) + 3^2 with 0 < 1 < 3.

%e a(14) = 1 since 14 = 1^2*(1+0^2+2^2) + 3^2 with 0 < 2 < 3.

%e a(15) = 1 since 15 = 1^2*(1+1^2+2^2) + 3^2 with 1 < 2 < 3.

%e a(23) = 1 since 23 = 1^2*(1+2^2+3^2) + 3^2 with 2 < 3 = 3.

%e a(38) = 1 since 38 = 1^2*(1+0^2+1^2) + 6^2 with 0 < 1 < 6.

%e a(47) = 1 since 47 = 1^2*(1+1^2+3^2) + 6^2 with 1 < 3 < 6.

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

%e a(77) = 1 since 77 = 1^2*(1+2^2+6^2) + 6^2 with 2 < 6 = 6.

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

%e a(152) = 1 since 152 = 2^2*(1+0^2+1^2) + 12^2 with 0 < 1 < 12.

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

%e a(608) = 1 since 608 = 4^2*(1+0^2+1^2) + 24^2 with 0 < 1 < 24.

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

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

%Y Cf. A000118, A000290, A271518, A275656.

%K nonn

%O 1,4

%A _Zhi-Wei Sun_, Aug 04 2016