cp's OEIS Frontend

A338121 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.

%I A338121 #18 Oct 11 2023 12:34:15
%S A338121 1,2,3,4,5,7,31,43,67,79,85,87,103,115,475,643,1015,1399,1495,1723,
%T A338121 1819,1939,1987
%N A338121 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 A338121 Conjecture: The sequence only has 23 terms as listed.
%C A338121 See also the related sequence A338094.
%H A338121 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 A338121 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 A338121 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 A338121 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 A338121 SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t A338121 FQ[n_]:=FQ[n]=n>1&&IntegerQ[Log[4,n]];
%t A338121 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 A338121 Cf. A000118, A000290, A000302, A338094, A338095, A338096, A338103, A338119.
%K A338121 nonn,more
%O A338121 1,2
%A A338121 _Zhi-Wei Sun_, Oct 11 2020