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%I #13 Jan 19 2021 21:01:46
%S 1,7,12,15,76,79,92,115,131,151,155,175,177,181,183,199,214,235,236,
%T 237,239,243,252,259,262,268,271,279,287,1351,1687,1693,1741,1867,
%U 2227,2557,2587,2671,2791,2803,2999,3031,3127,3207,3237,3587,3637,3646,3727,3815,3827,3853,3862,3980,4039,4141,4207,4221,4243,4319,4371,4381,4471,4497,4597,4607,4615,4627
%N Positive integers not congruent to 0 or 2 modulo 8 which cannot be written as x^2 + y^2 + z^2 + w^2 with x + 2*y + 3*z a positive power of 4, where x, y, z, w are nonnegative integers.
%C Conjecture: 4627 is the last term of this sequence.
%C This is equivalent to Conjecture 2 in A338096.
%C The sequence has no term after 4627 smaller than 5*10^6.
%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. If x,y,z,w are nonnegative integers with x^2 + y^2 + z^2 + w^2 = 1, then x, y, z, w are all smaller than 2, and x + 2*y + 3*z = 4^k for no positive integer k.
%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]==2,Goto[aa]];Do[If[SQ[m-x^2-y^2-z^2]&&FQ[x+2y+3z],Goto[aa]],{x,0,Sqrt[m]},{y,0,Sqrt[m-x^2]},{z,0,Sqrt[m-x^2-y^2]}];tab=Append[tab,m];Label[aa],{m,1,5000}];tab
%Y Cf. A000118, A000290, A000302, A338096.
%K nonn
%O 1,2
%A _Zhi-Wei Sun_, Oct 10 2020
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