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A350021 Number of ways to write n as w^4 + x^2 + y^2 + z^2 with x - y a power of two (including 2^0 = 1).

%I A350021 #14 Dec 09 2021 08:15:54
%S A350021 1,2,1,1,4,4,1,1,2,3,3,1,2,5,3,1,5,4,1,5,8,4,1,2,4,8,6,1,6,9,2,2,4,2,
%T A350021 6,7,4,4,2,3,9,11,4,2,7,5,1,1,2,8,8,4,5,5,1,5,9,4,5,4,5,8,4,1,8,10,3,
%U A350021 6,7,5,2,3,2,6,9,3,8,9,1,4,9,5,8,9,7,11,5,1,8,13,9,4,4,6,6,4,5,9,7,6
%N A350021 Number of ways to write n as w^4 + x^2 + y^2 + z^2 with x - y a power of two (including 2^0 = 1).
%C A350021 Conjecture: a(n) > 0 for all n > 0.
%C A350021 This is a new refinement of Lagrange's four-square theorem, and we have verified it for n up to 10^6.
%C A350021 If x - y = 2^k, then x^2 + y^2 = ((x+y)^2 + (2^k)^2)/2 and x + y >= 2^k. So the above conjecture implies the conjecture in A349661.
%C A350021 In his 2017 JNT paper, the author proved that each n = 0,1,2,... can be written as w^4 + x^2 + y^2 + z^2 with w,x,y,z nonnegative integers.
%C A350021 In his 2019 IJNT paper, the author proved that any positive integer can be written as w^2 + x^2 + y^2 + z^2 with w,x,y,z nonnegative integers such that x - y is a power of two (including 2^0 = 1).
%H A350021 Zhi-Wei Sun, <a href="/A350021/b350021.txt">Table of n, a(n) for n = 1..10000</a>
%H A350021 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 A350021 Zhi-Wei Sun, <a href="https://doi.org/10.1142/S1793042119501045">Restricted sums of four squares</a>, Int. J. Number Theory 15(2019), 1863-1893.  See also <a href="http://arxiv.org/abs/1701.05868">arXiv:1701.05868 [math.NT]</a>.
%H A350021 Zhi-Wei Sun, <a href="http://hitpress.hit.edu.cn/2021/1015/c12593a261001/page.htm">New Conjectures in Number Theory and Combinatorics</a> (in Chinese), Harbin Institute of Technology Press, 2021.
%e A350021 a(3) = 1 with 3 = 1^4 + 1^2 + 0^2 + 1^2 and 1 - 0 = 2^0.
%e A350021 a(4) = 1 with 4 = 0^4 + 2^2 + 0^2 + 0^2 and  2 - 0 = 2^1.
%e A350021 a(7) = 1 with 7 = 1^4 + 2^2 + 1^2 + 1^2 and 2 - 1 = 2^0.
%e A350021 a(8) = 1 with 8 = 0^4 + 2^2 + 0^2 + 2^2 and 2 - 0 = 2^1.
%e A350021 a(12) = 1 with 12 = 1^4 + 3^2 + 1^2 + 1^2 and 3 - 1 = 2^1.
%e A350021 a(19) = 1 with 19 = 0^4 + 3^2 + 1^2 + 3^2 and 3 - 1 = 2^1.
%e A350021 a(28) = 1 with 28 = 1^4 + 5^2 + 1^2 + 1^2 and 5 - 1 = 2^2.
%e A350021 a(47) = 1 with 47 = 1^4 + 3^2 + 1^2 + 6^2 and 3 - 1 = 2^1.
%e A350021 a(55) = 1 with 55 = 1^4 + 2^2 + 1^2 + 7^2 and 2 - 1 = 2^0.
%e A350021 a(88) = 1 with 88 = 0^4 + 6^2 + 4^2 + 6^2 and 6 - 4 = 2^1.
%e A350021 a(103) = 1 with 103 = 3^4 + 3^2 + 2^2 + 3^2 and 3 - 2 = 2^0.
%e A350021 a(193) = 1 with 193 = 2^4 + 8^2 + 7^2 + 8^2 and 8 - 7 = 2^0.
%e A350021 a(439) = 1 with 439 = 3^4 + 5^2 + 3^2 + 18^2 and 5 - 3 = 2^1.
%t A350021 SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t A350021 PowQ[n_]:=PowQ[n]=IntegerQ[Log[2,n]];
%t A350021 tab={};Do[r=0;Do[If[SQ[n-w^4-x^2-y^2]&&PowQ[y-x],r=r+1],{w,0,(n-1)^(1/4)},{x,0,Sqrt[(n-w^4)/2]},{y,x+1,Sqrt[n-w^4-x^2]}];tab=Append[tab,r],{n,1,100}];Print[tab]
%Y A350021 Cf. A000079, A000118, A000290, A000583, A279612, A300708, A338094, A349661, A350012.
%K A350021 nonn
%O A350021 1,2
%A A350021 _Zhi-Wei Sun_, Dec 08 2021