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