A301479 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.

1, 53, 69, 71, 77, 87, 101, 103, 106, 117, 127, 133, 138, 142, 149, 159, 173, 174, 181, 191, 197, 199, 202, 206, 207, 212, 213, 221, 223, 229, 231, 234, 266, 269, 276, 277, 284, 293, 298, 309, 311, 325, 341, 346, 348, 351, 357, 362, 365, 373, 389, 398, 404, 412, 423, 424, 426, 429

Offset: 1

Author

Zhi-Wei Sun, Mar 22 2018

Keywords

• nonn

Comments

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.

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).

Examples

			a(1) = 1 since x^2 + 2*y^2 + 3*2^z > 1^3 for all x,y,z = 0,1,2,....
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000079, A000578, A002479, A301471, A301472.

Programs

Mathematica

f[n_]:=f[n]=FactorInteger[n];
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;
QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
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]