A338096 Number of ways to write 2*n+1 as x^2 + y^2 + z^2 + w^2 with x + 2*y + 3*z a positive power of two, where x, y, z, w are nonnegative integers.

1, 1, 5, 1, 3, 2, 3, 2, 5, 1, 5, 2, 4, 4, 7, 2, 5, 5, 3, 3, 6, 1, 5, 3, 2, 6, 6, 2, 4, 2, 2, 2, 8, 2, 7, 3, 5, 6, 6, 1, 5, 6, 7, 7, 8, 4, 6, 5, 5, 7, 11, 3, 13, 5, 3, 6, 11, 4, 7, 6, 3, 7, 9, 5, 8, 6, 3, 8, 9, 5, 10, 3, 9, 8, 7, 2, 7, 6, 5, 4, 4, 3, 12, 7, 3, 9, 9, 5, 11, 8, 2, 5, 10, 3, 5, 5, 2, 9, 9, 4, 13

Offset: 0

Author

Zhi-Wei Sun, Oct 09 2020

Keywords

• nonn

Comments

Conjecture 1 (1-2-3 Conjecture): a(n) > 0 for all n >= 0. In other words, any positive odd integer m can be written as x^2 + y^2 + z^2 + w^2 with x, y, z, w nonnegative integers such that x + 2*y + 3*z = 2^k for some positive integer k.

Conjecture 2 (Strong Version of the 1-2-3 Conjecture): For any integer m > 4627 not congruent to 0 or 2 modulo 8, we can write m as x^2 + y^2 + z^2 + w^2 with x, y, z, w nonnegative integers such that x + 2*y + 3*z = 4^k for some positive integer k.

We have verified Conjectures 1 and 2 for m up to 5*10^6. Conjecture 2 implies that A299924(n) > 0 for all n > 0.

By Theorem 1.2(v) of the author's 2017 JNT paper, any positive integer n can be written as x^2 + y^2 + z^2 + 4^k with k, x, y, z nonnegative integers.

See also A338094 and A338095 for similar conjectures.

Examples

			a(1) = 1, and 2*1 + 1 = 1^2 + 0^2 + 1^2 + 1^2 with 1 + 2*0 + 3*1 = 2^2.
a(3) = 1, and 2*3 + 1 = 1^2 + 2^2 + 1^2 + 1^2 with 1 + 2*2 + 3*1 = 2^3.
a(9) = 1, and 2*9 + 1 = 1^2 + 6^2 + 1^2 + 1^2 with 1 + 2*6 + 3*1 = 2^4.
a(21) = 1, and 2*21 + 1 = 5^2 + 4^2 + 1^2 + 1^2 with 5 + 2*4 + 3*1 = 2^4.
a(39) = 1, and 2*39 + 1 = 1^2 + 5^2 + 7^2 + 2^2 with 1 + 2*5 + 3*7 = 2^5.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
• 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. A000079, A000118, A000290, A000302, A279612, A299924, A338094, A338095, A338103, A338119, A338121.

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
PQ[n_]:=PQ[n]=n>1&&IntegerQ[Log[2,n]];
tab={};Do[r=0;Do[If[SQ[2n+1-x^2-y^2-z^2]&&PQ[x+2y+3z],r=r+1],{x,0,Sqrt[2n+1]},{y,Boole[x==0],Sqrt[2n+1-x^2]},{z,0,Sqrt[2n+1-x^2-y^2]}]; tab=Append[tab,r],{n,0,100}];Print[tab]