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

1,2

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.

Links

Table of n, a(n) for n=1..68.

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.

Example

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.

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

Crossrefs

Cf. A000118, A000290, A000302, A338096.

Sequence in context: A071780 A353219 A305553 * A190720 A295625 A063303

Adjacent sequences: A338100 A338101 A338102 * A338104 A338105 A338106

Keyword

nonn

Author

Zhi-Wei Sun, Oct 10 2020

Status

approved