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

1,2

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).

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.

Example

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,....

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]

Crossrefs

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

Sequence in context: A087287 A349066 A266877 * A041721 A048358 A124456

Adjacent sequences: A301469 A301470 A301471 * A301473 A301474 A301475

Keyword

nonn

Author

Zhi-Wei Sun, Mar 21 2018

Status

approved