A338121 Positive integers not congruent to 0 mod 6 which cannot be written as x^2 + y^2 + z^2 + w^2 with x + y = 4^k for some positive integer k, where x, y, z, w are nonnegative integers.

1, 2, 3, 4, 5, 7, 31, 43, 67, 79, 85, 87, 103, 115, 475, 643, 1015, 1399, 1495, 1723, 1819, 1939, 1987

Offset: 1

Author

Zhi-Wei Sun, Oct 11 2020

Keywords

• nonn
• more

Comments

Conjecture: The sequence only has 23 terms as listed.

See also the related sequence A338094.

Examples

			a(n) = n for n = 1..5, this is because x + y < 4 if x, y, z, w are nonnegative integers satisfying x^2 + y^2 + z^2 + w^2 <= 5.
		

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, A338094, A338095, A338096, A338103, A338119.

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]==6,Goto[aa]];Do[If[SQ[m-x^2-y^2-z^2]&&FQ[x+y],Goto[aa]],{x,0,Sqrt[m/2]},{y,x,Sqrt[m-x^2]},{z,0,Sqrt[(m-x^2-y^2)/2]}];tab=Append[tab,m];Label[aa],{m,1,2000}];tab