cp's OEIS Frontend

A338139 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x^2 + 26*y^2 - 11*x*y a power of two (including 2^0 = 1), where x, y, z, w are nonnegative integers with z <= w.

%I A338139 #13 Jan 19 2021 21:02:10
%S A338139 1,2,2,2,3,4,2,2,4,5,3,4,3,4,3,2,4,6,3,5,6,4,2,4,4,5,4,4,4,6,2,2,7,5,
%T A338139 3,6,5,4,3,5,7,8,1,4,8,4,2,4,5,6,4,5,5,6,4,4,8,5,2,6,4,3,3,2,8,11,3,5,
%U A338139 11,6,1,6,8,7,5,4,6,5,1,5,10,10,5,9,8,5,4,4,8,14,5,5,8,4,4,4,6,7,5,7
%N A338139 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x^2 + 26*y^2 - 11*x*y a power of two (including 2^0 = 1), where x, y, z, w are nonnegative integers with z <= w.
%C A338139 Conjecture: a(n) > 0 for all n > 0. Moreover, any positive integer n congruent to 1 or 2 modulo 4 can be written as x^2 + y^2 + z^2 + w^2 with x, y, z, w nonnegative integers such that x^2 + 26*y^2 - 11*x*y = 4^k for some nonnegative integer k.
%C A338139 We have verified this for all n = 1..10^8.
%C A338139 See also A337082 for a similar conjecture.
%H A338139 Zhi-Wei Sun, <a href="/A338139/b338139.txt">Table of n, a(n) for n = 1..10000</a>
%H A338139 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 A338139 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 A338139 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 A338139 a(1) = 1, and 1 = 1^2 + 0^2 + 0^2 + 0^2 with 1^2 + 26*0^2 - 11*1*0 = 2^0.
%e A338139 a(43) = 1, and 43 = 1^2 + 1^2 + 4^2 + 5^2 with 1^2 + 26*1^2 - 11*1*1 = 2^4.
%e A338139 a(6547) = 1, and 6547 = 17^2 + 1^2 + 4^2 + 79^2 with 17^2 + 26*1^2 - 11*17*1 = 2^7.
%e A338139 a(11843) = 1, and 11843 = 3^2 + 1^2 + 13^2 + 108^2 with 3^2 + 26*1^2 - 11*3*1 = 2^1.
%t A338139 SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t A338139 PQ[n_]:=PQ[n]=n>0&&IntegerQ[Log[2,n]];
%t A338139 tab={};Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&PQ[x^2+26*y^2-11*x*y],r=r+1],{x,0,Sqrt[n]},{y,Boole[x==0],Sqrt[n-x^2]},{z,0,Sqrt[(n-x^2-y^2)/2]}];tab=Append[tab,r],{n,1,100}];tab
%Y A338139 Cf. A000079, A000118, A000290, A279612, A282463, A338094, A338095, A338096, A338103, A338119, A338121.
%K A338139 nonn
%O A338139 1,2
%A A338139 _Zhi-Wei Sun_, Oct 12 2020