A338103 Positive integers not congruent to 0 or 2 modulo 8 which cannot be written as x^2 + y^2 + z^2 + w^2 with x + 2*y + 3*z a positive power of 4, where x, y, z, w are nonnegative integers.

1, 7, 12, 15, 76, 79, 92, 115, 131, 151, 155, 175, 177, 181, 183, 199, 214, 235, 236, 237, 239, 243, 252, 259, 262, 268, 271, 279, 287, 1351, 1687, 1693, 1741, 1867, 2227, 2557, 2587, 2671, 2791, 2803, 2999, 3031, 3127, 3207, 3237, 3587, 3637, 3646, 3727, 3815, 3827, 3853, 3862, 3980, 4039, 4141, 4207, 4221, 4243, 4319, 4371, 4381, 4471, 4497, 4597, 4607, 4615, 4627

Offset: 1

Author

Zhi-Wei Sun, Oct 10 2020

Keywords

• nonn

Comments

Conjecture: 4627 is the last term of this sequence.

This is equivalent to Conjecture 2 in A338096.

The sequence has no term after 4627 smaller than 5*10^6.

Examples

			a(1) = 1. If x,y,z,w are nonnegative integers with x^2 + y^2 + z^2 + w^2 = 1, then x, y, z, w are all smaller than 2, and x + 2*y + 3*z = 4^k for no positive integer k.
		

Links

• Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190. See also arXiv:1604.06723 [math.NT].
• Zhi-Wei Sun, Restricted sums of four squares, Int. J. Number Theory 15(2019), 1863-1893. See also arXiv:1701.05868 [math.NT].
• Zhi-Wei Sun, Sums of four squares with certain restrictions, arXiv:2010.05775 [math.NT], 2020.

Crossrefs

Cf. A000118, A000290, A000302, A338096.

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
FQ[n_]:=FQ[n]=n>1&&IntegerQ[Log[4,n]];
tab={};Do[If[Mod[m,8]==0||Mod[m,8]==2,Goto[aa]];Do[If[SQ[m-x^2-y^2-z^2]&&FQ[x+2y+3z],Goto[aa]],{x,0,Sqrt[m]},{y,0,Sqrt[m-x^2]},{z,0,Sqrt[m-x^2-y^2]}];tab=Append[tab,m];Label[aa],{m,1,5000}];tab