|
%I #16 Feb 11 2017 11:22:06
%S 1,2,2,2,2,2,2,1,1,3,3,2,2,2,2,1,1,3,4,3,2,2,2,1,1,3,4,4,2,2,3,1,2,4,
%T 5,4,4,4,4,2,2,6,5,3,3,4,4,1,2,5,7,6,4,4,6,3,2,5,5,5,2,4,5,2,2,6,8,5,
%U 5,5,5,1,3,7,6,6,4,5,4,1,2
%N Number of ordered ways to write n as w^5 + x^2 + y^2 + z^2, where w,x,y,z are nonnegative integers with x <= y <= z.
%C Let c be 1 or 4. Then any nonnegative integer n can be written as c*w^5 + x^2 + y^2 + z^2 with w,x,y,z nonnegative integers. We now prove this by induction on n. For n < 2^(10) this can be verified directly via a computer. If 2^(10) divides n, then by the induction hypothesis we can write n/2^(10) as c*w^5 + x^2 + y^2 + z^2 with w,x,y,z, nonnegative integers, and hence n = c*(2^2*w)^5 + (2^5*x)^2 + (2^5*y)^2 + (2^5*z)^2. If n is not of the form 4^k*(8m+7) with k and m nonnegative integers, then n is the sum of three squares and hence n = c*0^5 + x^2 + y^2 + z^2 for some integers x,y,z. When n = 4^k*(8m+7) > 2^(10) with k < 5, it is easy to see that n - c*1^5 or n - c*2^5 is the sum of three squares.
%C For any positive integer k and for each c = 2, 6, any natural number n can be written as c*w^k + x^2 + y^2 + z^2 with w,x,y,z nonnegative integers. In fact, for every n = 0,1,2,... either n - c*0^k or n - c*1^k can be written as the sum of three squares.
%C See also A270969 and A273429 for similar results.
%C For some conjectural refinements of Lagrange's four-square theorem, one may consult the author's preprint arXiv:1604.06723
%H Zhi-Wei Sun, <a href="/A273915/b273915.txt">Table of n, a(n) for n = 0..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.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.
%e a(0) = 1 since 0 = 0^5 + 0^2 + 0^2 + 0^2.
%e a(7) = 1 since 7 = 1^5 + 1^2 + 1^2 + 2^2.
%e a(8) = 1 since 8 = 0^5 + 0^2 + 2^2 + 2^2.
%e a(15) = 1 since 15 = 1^5 + 1^2 + 2^2 + 3^2.
%e a(16) = 1 since 16 = 0^5 + 0^2 + 0^2 + 4^2.
%e a(23) = 1 since 23 = 1^5 + 2^2 + 3^2 + 3^2.
%e a(24) = 1 since 24 = 0^2 + 2^2 + 2^2 + 4^2.
%e a(31) = 1 since 31 = 1^5 + 1^2 + 2^2 + 5^2.
%e a(47) = 1 since 47 = 1^5 + 1^2 + 3^2 + 6^2.
%e a(71) = 1 since 71 = 1^5 + 3^2 + 5^2 + 6^2.
%e a(79) = 1 since 79 = 1^5 + 2^2 + 5^2 + 7^2.
%e a(92) = 1 since 92 = 1^5 + 1^2 + 3^2 + 9^2.
%e a(112) = 1 since 112 = 2^5 + 0^2 + 4^2 + 8^2.
%e a(143) = 1 since 143 = 1^5 + 5^2 + 6^2 + 9^2.
%e a(191) = 1 since 191 = 1^5 + 3^2 + 9^2 + 10^2.
%e a(240) = 1 since 240 = 2^5 + 0^2 + 8^2 + 12^2.
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
%t Do[r=0;Do[If[SQ[n-w^5-x^2-y^2],r=r+1],{w,0,n^(1/5)},{x,0,Sqrt[(n-w^5)/3]},{y,x,Sqrt[(n-w^5-x^2)/2]}];Print[n," ",r];Label[aa];Continue,{n,0,80}]
%Y Cf. A000118, A000290, A000584, A270969, A273429, A273917.
%K nonn
%O 0,2
%A _Zhi-Wei Sun_, Jun 03 2016
|
