login
Positive integers not congruent to 0 mod 6 which cannot be written as x^2 + y^2 + z^2 + w^2 with x + y = 4^k for some positive integer k, where x, y, z, w are nonnegative integers.
7

%I #18 Oct 11 2023 12:34:15

%S 1,2,3,4,5,7,31,43,67,79,85,87,103,115,475,643,1015,1399,1495,1723,

%T 1819,1939,1987

%N Positive integers not congruent to 0 mod 6 which cannot be written as x^2 + y^2 + z^2 + w^2 with x + y = 4^k for some positive integer k, where x, y, z, w are nonnegative integers.

%C Conjecture: The sequence only has 23 terms as listed.

%C See also the related sequence A338094.

%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(n) = n for n = 1..5, this is because x + y < 4 if x, y, z, w are nonnegative integers satisfying x^2 + y^2 + z^2 + w^2 <= 5.

%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];

%t FQ[n_]:=FQ[n]=n>1&&IntegerQ[Log[4,n]];

%t tab={};Do[If[Mod[m,8]==0||Mod[m,8]==6,Goto[aa]];Do[If[SQ[m-x^2-y^2-z^2]&&FQ[x+y],Goto[aa]],{x,0,Sqrt[m/2]},{y,x,Sqrt[m-x^2]},{z,0,Sqrt[(m-x^2-y^2)/2]}];tab=Append[tab,m];Label[aa],{m,1,2000}];tab

%Y Cf. A000118, A000290, A000302, A338094, A338095, A338096, A338103, A338119.

%K nonn,more

%O 1,2

%A _Zhi-Wei Sun_, Oct 11 2020