cp's OEIS Frontend

A268197 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with w*(25*w + 24*x + 48*y + 96*z) a square, where w is a positive integer and x,y,z are nonnegative integers.

%I A268197 #37 May 05 2016 02:41:10
%S A268197 1,2,1,1,2,2,1,2,3,2,2,3,3,3,1,1,4,5,2,2,3,4,1,2,2,4,8,3,4,4,1,2,5,1,
%T A268197 5,4,2,7,3,2,6,7,1,4,7,7,3,3,8,5,4,5,6,6,1,3,8,3,6,3,2,8,5,1,5,6,5,7,
%U A268197 6,6
%N A268197 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with w*(25*w + 24*x + 48*y + 96*z) a square, where w is a positive integer and x,y,z are nonnegative integers.
%C A268197 Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 3, 7, 15, 23, 43, 55, 463, 4^k*m (k = 0,1,2,... and m = 1, 31, 34).
%C A268197 (ii) For each triple (a,b,c) = (1,3,4), (2,3,4), (2,4,6), any positive integer can be written as w^2 + x^2 + y^2 + z^2 with w*(25*w + 24*(a*x+b*y+c*z)) a square, where w is a positive integer and x,y,z are nonnegative integers.
%C A268197 For more refinements of Lagrange's four-square theorem, see arXiv:1604.06723.
%H A268197 Zhi-Wei Sun, <a href="/A268197/b268197.txt">Table of n, a(n) for n = 1..10000</a>
%H A268197 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1604.06723">Refining Lagrange's four-square theorem</a>, arXiv:1604.06723 [math.GM], 2016.
%H A268197 Zhi-Wei Sun, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;852b9c4a.1604">Refine Lagrange's four-square theorem</a>, a message to Number Theory List, April 26, 2016.
%e A268197 a(1) = 1 since 1 = 1^2 + 0^2 + 0^2 + 0^2 with 1 > 0 and 1*(25*1 + 24*0 + 48*0 + 96*0) = 5^2.
%e A268197 a(2) = 2 since 2 = 1^2 + 0^2 + 0^2 + 1^2 with 1 > 0 and 1*(25*1 + 24*0 + 48*0 + 96*1) = 11^2, and also 2 = 1^2 + 1^2 + 0^2 + 0^2 with 1 > 0 and 1*(25*1 + 24*1 + 48*0 + 96*0) = 7^2.
%e A268197 a(3) = 1 since 3 = 1^2 + 0^2 + 1^2 + 1^2 with 1 > 0 and 1*(25*1 + 24*0 + 48*1 + 96*1) = 13^2.
%e A268197 a(7) = 1 since 7 = 1^2 + 1^2 + 1^2 + 2^2 with 1 > 0 and 1*(25*1 + 24*1 + 48*1 + 96*2) = 17^2.
%e A268197 a(15) = 1 since 15 = 1^2 + 3^2 + 2^2 + 1^2 with 1 > 0 and 1*(25*1 + 24*3 + 48*2 + 96*1) = 17^2.
%e A268197 a(23) = 1 since 23 = 3^2 + 2^2 + 3^2 + 1^2 with 3 > 0 and 3*(25*3 + 24*2 + 48*3 + 96*1) = 33^2.
%e A268197 a(31) = 1 since 31 = 1^2 + 1^2 + 2^2 + 5^2 with 1 > 0 and 1*(25*1 + 24*1 + 48*2 + 96*5) = 25^2.
%e A268197 a(34) = 1 since 34 = 1^2 + 1^2 + 4^2 + 4^2 with 1 > 0 and 1*(25*1 + 24*1 + 48*4 + 96*4) = 25^2.
%e A268197 a(43) = 1 since 43 = 3^2 + 3^2 + 3^2 + 4^2 with 3 > 0 and 3*(25*3 + 24*3 + 48*3 + 96*4) = 45^2.
%e A268197 a(55) = 1 since 55 = 3^2 + 1^2 + 6^2 + 3^2 with 3 > 0 and 3*(25*3 + 24*1 + 48*6 + 96*3) = 45^2.
%e A268197 a(463) = 1 since 463 = 3^2 + 18^2 + 11^2 + 3^2 with 3 > 0 and 3*(25*3 + 24*18 + 48*11 + 96*3) = 63^2.
%t A268197 SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
%t A268197 Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&SQ[25x^2+24x(y+2z+4*Sqrt[n-x^2-y^2-z^2])],r=r+1],{x,1,Sqrt[n]},{y,0,Sqrt[n-x^2]},{z,0,Sqrt[n-x^2-y^2]}];Print[n," ",r];Continue,{n,1,70}]
%Y A268197 Cf. A000118, A000290, A260625, A261876, A262357, A267121, A268507, A269400, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824, A272084, A272332, A272351, A272620.
%K A268197 nonn
%O A268197 1,2
%A A268197 _Zhi-Wei Sun_, May 04 2016