|
%I #16 May 25 2023 09:17:16
%S 1,1,2,2,1,1,4,1,1,3,1,3,2,1,3,3,2,3,5,2,3,4,6,1,3,5,1,6,1,3,7,2,2,5,
%T 6,5,6,3,6,4,1,3,4,5,4,5,7,2,3,8,6,7,3,4,8,3,2,6,3,5,7,3,8,7,2,4,10,4,
%U 4,7,9,7,2,4,2,7,3,5,11,2,4
%N Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and y <= z such that 2*x + y - z is either zero or a power of 8 (including 8^0 = 1).
%C Conjecture: (i) For any c = 1,2,4, each n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and y <= z such that c*(2*x+y-z) is either zero or a power of eight (including 8^0 = 1).
%C (ii) Each n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that P(x,y,z,w) is either zero or a power of four (including 4^0 = 1), whenever P(x,y,z,w) is among the polynomials 2*x-y, x+y-z, x-y-z, x+y-2*z, 2*x+y-z, 2*x-y-z, 2*x-2*y-z, x+2*y-3*z, 2*x+2*y-2*z, 2*x+2*y-4*z, 3*x-2*y-z, x+3*y-3*z, 2*x+3*y-3*z, 4*x+2*y-2*z, 8*x+2*y-2*z, 2*(x-y)+z-w, 4*(x-y)+2*(z-w).
%C Part (i) of the conjecture is stronger than the first part of Conjecture 4.4 in the linked JNT paper (see also A273432).
%C Modifying the proofs of Theorem 1.1 and Theorem 1.2(i) in the linked JNT paper slightly, we see that for any a = 1,4 and m = 4,5,6 we can write each n = 0,1,2,... as a*x^m + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x is either zero or a power of two (including 2^0 = 1), and that for any b = 1,2 each n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that b*(x-y) is either zero or a power of 4 (including 4^0 = 1).
%C Starts to differ from A273432 at n=197. - _R. J. Mathar_, May 25 2023
%H Zhi-Wei Sun, <a href="/A284343/b284343.txt">Table of n, a(n) for n = 0..10000</a>
%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.
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1701.05868">Restricted sums of four squares</a>, arXiv:1701.05868 [math.NT], 2017.
%e a(4) = 1 since 4 = 0^2 + 0^2 + 0^2 + 2^2 with 0 = 0 and 2*0 + 0 - 0 = 0.
%e a(5) = 1 since 5 = 1^2 + 0^2 + 2^2 + 0^2 with 0 < 2 and 2*1 + 0 - 2 = 0.
%e a(7) = 1 since 7 = 1^2 + 1^2 + 2^2 + 1^2 with 1 < 2 and 2*1 + 1 - 2 = 8^0.
%e a(40) = 1 since 40 = 4^2 + 2^2 + 2^2 + 4^2 with 2 = 2 and 2*4 + 2 - 2 = 8.
%e a(138) = 1 since 138 = 3^2 + 5^2 + 10^2 + 2^2 with 5 < 10 and 2*3 + 5 - 10 = 8^0.
%e a(1832) = 1 since 1832 = 4^2 + 30^2 + 30^2 + 4^2 with 30 = 30 and 2*4 + 30 - 30 = 8.
%e a(2976) = 1 since 2976 = 20^2 + 16^2 + 48^2 + 4^2 with 16 < 48 and 2*20 + 16 - 48 = 8.
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t Pow[n_]:=Pow[n]=n==0||(n>0&&IntegerQ[Log[8,n]]);
%t Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&Pow[2x+y-z],r=r+1],{x,0,Sqrt[n]},{y,0,Sqrt[(n-x^2)/2]},{z,y,Sqrt[n-x^2-y^2]}];Print[n," ",r],{n,0,80}]
%Y Cf. A000079, A000118, A000290, A000302, A000578, A001018, A271518, A273432, A279612.
%K nonn
%O 0,3
%A _Zhi-Wei Sun_, Mar 25 2017
|
