A301472 Positive integers not of the form x^2 + 2*y^2 + 3*2^z with x,y,z nonnegative integers.

1, 2, 77, 154, 157, 173, 285, 308, 311, 314, 317, 346, 383, 397, 477, 493, 509, 557, 570, 616, 621, 634, 692, 701, 717, 727, 733, 757, 766, 794, 797, 877, 909, 954, 957, 986, 997, 1013, 1018, 1069, 1085, 1093, 1111, 1114, 1117, 1181, 1197, 1221, 1232, 1242, 1268, 1277, 1293

Offset: 1

Author

Zhi-Wei Sun, Mar 21 2018

Keywords

• nonn

Comments

It might seem that 1 is the only square in this sequence, but 5884015571^2 is also a term of this sequence.

See also A301471 for related information.

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

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
• Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.

Crossrefs

Cf. A000079, A000290, A002479, A299924, A299537, A299794, A300219, A300362, A300396, A300510, A301376, A301391, A301452, A301471, A301479.

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*2^k],Goto[aa]],{k,0,Log[2,m/3]}];tab=Append[tab,m];Label[aa],{m,1,1293}];Print[tab]