A338119 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 + y + 2*z a positive power of 4, where x, y, z, w are nonnegative integers.

1, 15, 22, 23, 27, 31, 36, 37, 38, 183, 193, 223, 237, 254, 279, 283, 285, 310, 311, 325, 331, 343, 349, 358, 359, 379, 381, 389, 399, 421, 429, 430, 436, 447, 463, 465, 471, 475, 479, 483, 503, 511, 513, 516, 523, 541, 547, 553, 555, 556, 557, 559, 563, 565, 566, 598, 599, 603, 604, 611, 625, 631, 639, 645, 647, 649, 651

Offset

1,2

Comments

Conjecture: The sequence has exactly 124 terms as listed in the b-file with 10839 the last one.

See also the related sequence A338095.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..124

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, for, if x, y, z, w are nonnegative integers with x^2 + y^2 + z^2 + w^2 = 1 then x + y + 2*z < 4.

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+y+2z], Goto[aa]], {x, 0, Sqrt[m/2]}, {y, x, Sqrt[m-x^2]}, {z, 0, Sqrt[m-x^2-y^2]}]; tab=Append[tab, m]; Label[aa], {m, 1, 660}]; tab

Crossrefs

Cf. A000118, A000290, A000302, A338094, A338095, A338096, A338103, A338121.

Sequence in context: A236764 A300958 A066758 * A297931 A018888 A115174

Adjacent sequences: A338116 A338117 A338118 * A338120 A338121 A338122

Keyword

nonn

Author

Zhi-Wei Sun, Oct 10 2020

Status

approved