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%I #22 Apr 15 2018 12:20:14
%S 1,2,77,154,157,173,285,308,311,314,317,346,383,397,477,493,509,557,
%T 570,616,621,634,692,701,717,727,733,757,766,794,797,877,909,954,957,
%U 986,997,1013,1018,1069,1085,1093,1111,1114,1117,1181,1197,1221,1232,1242,1268,1277,1293
%N Positive integers not of the form x^2 + 2*y^2 + 3*2^z with x,y,z nonnegative integers.
%C It might seem that 1 is the only square in this sequence, but 5884015571^2 is also a term of this sequence.
%C See also A301471 for related information.
%C It is known that a positive integer n has the form x^2 + 2*y^2 with x and y integers if and only if the p-adic order of n is even for any prime p == 5 or 7 (mod 8).
%H Zhi-Wei Sun, <a href="/A301472/b301472.txt">Table of n, a(n) for n = 1..10000</a>
%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.
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1701.05868">Restricted sums of four squares</a>, arXiv:1701.05868 [math.NT], 2017-2018.
%e a(1) = 1 and a(2) = 2 since x^2 + 2*y^2 + 3*2^z > 2 for all x,y,z = 0,1,2,....
%t f[n_]:=f[n]=FactorInteger[n];
%t g[n_]:=g[n]=Sum[Boole[(Mod[Part[Part[f[n],i],1],8]==5||Mod[Part[Part[f[n],i],1],8]==7)&&Mod[Part[Part[f[n],i],2],2]==1],{i,1,Length[f[n]]}]==0;
%t QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t tab={};Do[Do[If[QQ[m-3*2^k],Goto[aa]],{k,0,Log[2,m/3]}];tab=Append[tab,m];Label[aa],{m,1,1293}];Print[tab]
%Y Cf. A000079, A000290, A002479, A299924, A299537, A299794, A300219, A300362, A300396, A300510, A301376, A301391, A301452, A301471, A301479.
%K nonn
%O 1,2
%A _Zhi-Wei Sun_, Mar 21 2018
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