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

Author

Zhi-Wei Sun, Oct 10 2020

Keywords

• nonn

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.

Examples

			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.
		

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.

Crossrefs

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

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]==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