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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.
7
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
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
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Oct 10 2020
STATUS
approved