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%I #24 Jan 19 2021 21:01:25
%S 1,1,1,2,2,2,2,3,2,3,1,2,3,3,1,5,3,2,3,5,2,5,3,4,4,4,3,6,4,3,4,5,3,7,
%T 2,4,6,5,2,6,3,3,4,7,3,6,4,4,5,5,2,7,2,2,3,5,4,6,4,4,4,6,3,9,4,5,6,5,
%U 3,7,2,5,7,7,4,10,7,6,7,9,3,8,3,4,7,7,5,10,6,5,6,10,6,11,5,5,9,5,3,12
%N Number of ways to write 2*n + 1 as x^2 + y^2 + z^2 + w^2 with x + y a positive power of two, where x, y, z, w are nonnegative integers with x <= y and z <= w.
%C Conjecture: a(n) > 0 for all n > 0. Moreover, any integer m > 1987 not congruent to 0 or 6 modulo 8 can be written as x^2 + y^2 + z^2 + w^2 with x, y, z, w nonnegative integers and x + y a positive power of 4.
%C We have verified the latter version of the conjecture for m up to 3*10^7.
%C By Theorem 1.1(ii) of the author's IJNT paper, any positive integer can be written as x^2 + y^2 + z^2 + w^2 with x, y, z, w nonnegative integers and x - y a power of two (including 2^0 = 1).
%C See also A338121 for related information, and A338095 and A338096 for similar conjectures.
%H Zhi-Wei Sun, <a href="/A338094/b338094.txt">Table of n, a(n) for n = 1..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. See also <a href="http://arxiv.org/abs/1604.06723">arXiv:1604.06723 [math.NT]</a>.
%H 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 Zhi-Wei Sun, <a href="https://arxiv.org/abs/2010.05775">Sums of four squares with certain restrictions</a>, arXiv:2010.05775 [math.NT], 2020.
%e a(1) = 1, and 2*1 + 1 = 1^2 + 1^2 + 0^2 + 1^2 with 1 + 1 = 2^1.
%e a(2) = 1, and 2*2 + 1 = 0^2 + 2^2 + 0^2 + 1^2 with 0 + 2 = 2^1.
%e a(3) = 1, and 2*3 + 1 = 1^2 + 1^2 + 1^2 + 2^2 with 1 + 1 = 2^1.
%e a(11) = 1, and 2*11 + 1 = 1^2 + 3^2 + 2^2 + 3^2 with 1 + 3 = 2^2.
%e a(15) = 1, and 2*15 + 1 = 1^2 + 1^2 + 2^2 + 5^2 with 1 + 1 = 2^1.
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t PQ[n_]:=PQ[n]=n>1&&IntegerQ[Log[2,n]];
%t tab={};Do[r=0;Do[If[SQ[2n+1-x^2-y^2-z^2]&&PQ[x+y],r=r+1],{x,0,Sqrt[(2n+1)/2]},{y,x,Sqrt[2n+1-x^2]},{z,Boole[x+y==0],Sqrt[(2n+1-x^2-y^2)/2]}];
%t tab=Append[tab,r],{n,1,100}];Print[tab]
%Y Cf. A000079, A000118, A000290, A000302, A279612, A338095, A338096, A338103, A338103, A338119, A338121.
%K nonn
%O 1,4
%A _Zhi-Wei Sun_, Oct 09 2020
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