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%I #17 Mar 22 2018 09:46:37
%S 1,53,69,71,77,87,101,103,106,117,127,133,138,142,149,159,173,174,181,
%T 191,197,199,202,206,207,212,213,221,223,229,231,234,266,269,276,277,
%U 284,293,298,309,311,325,341,346,348,351,357,362,365,373,389,398,404,412,423,424,426,429
%N Positive integers m such that m^3 cannot be written in the form x^2 + 2*y^2 + 3*2^z with x,y,z nonnegative integers.
%C It seems that this sequence has infinitely many terms. In contrast, the author conjectured in A301471 and A301472 that any square greater than one can be written as x^2 + 2*y^2 + 3*2^z with x,y,z nonnegative integers.
%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="/A301479/b301479.txt">Table of n, a(n) for n = 1..10000</a>
%e a(1) = 1 since x^2 + 2*y^2 + 3*2^z > 1^3 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^3-3*2^k],Goto[aa]],{k,0,Log[2,m^3/3]}];tab=Append[tab,m];Label[aa],{m,1,429}];Print[tab]
%Y Cf. A000079, A000578, A002479, A301471, A301472.
%K nonn
%O 1,2
%A _Zhi-Wei Sun_, Mar 22 2018
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