cp's OEIS Frontend

A338119 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 + y + 2*z a positive power of 4, where x, y, z, w are nonnegative integers.

%I A338119 #13 Jan 19 2021 21:02:00
%S A338119 1,15,22,23,27,31,36,37,38,183,193,223,237,254,279,283,285,310,311,
%T A338119 325,331,343,349,358,359,379,381,389,399,421,429,430,436,447,463,465,
%U A338119 471,475,479,483,503,511,513,516,523,541,547,553,555,556,557,559,563,565,566,598,599,603,604,611,625,631,639,645,647,649,651
%N A338119 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 + y + 2*z a positive power of 4, where x, y, z, w are nonnegative integers.
%C A338119 Conjecture: The sequence has exactly 124 terms as listed in the b-file with 10839 the last one.
%C A338119 See also the related sequence A338095.
%H A338119 Zhi-Wei Sun, <a href="/A338119/b338119.txt">Table of n, a(n) for n = 1..124</a>
%H A338119 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 A338119 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 A338119 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 A338119 a(1) = 1, for, if x, y, z, w are nonnegative integers with x^2 + y^2 + z^2 + w^2 = 1 then x + y + 2*z < 4.
%t A338119 SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t A338119 FQ[n_]:=FQ[n]=n>1&&IntegerQ[Log[4,n]];
%t A338119 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+y+2z],Goto[aa]],{x,0,Sqrt[m/2]},{y,x,Sqrt[m-x^2]},{z,0,Sqrt[m-x^2-y^2]}];tab=Append[tab,m];Label[aa],{m,1,660}];tab
%Y A338119 Cf. A000118, A000290, A000302, A338094, A338095, A338096, A338103, A338121.
%K A338119 nonn
%O A338119 1,2
%A A338119 _Zhi-Wei Sun_, Oct 10 2020