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

1,2

Comments

Conjecture: The sequence only has 23 terms as listed.

See also the related sequence A338094.

Links

Table of n, a(n) for n=1..23.

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(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.

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

Crossrefs

Cf. A000118, A000290, A000302, A338094, A338095, A338096, A338103, A338119.

Sequence in context: A270831 A063738 A063737 * A278342 A276522 A135709

Adjacent sequences: A338118 A338119 A338120 * A338122 A338123 A338124

Keyword

nonn,more

Author

Zhi-Wei Sun, Oct 11 2020

Status

approved