A338096 - OEIS

%I #31 Jan 19 2021 21:01:38

%S 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,

%T 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,

%U 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

%N 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.

%C 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.

%C 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.

%C 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.

%C 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.

%C See also A338094 and A338095 for similar conjectures.

%H Zhi-Wei Sun, <a href="/A338096/b338096.txt">Table of n, a(n) for n = 0..10000</a>

%H Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2016.11.008">Refining Lagrange's four-square theorem</a>, J. Number Theory 175(2017), 167-190. See also <a href="http://arxiv.org/abs/1604.06723">arXiv:1604.06723 [math.NT]</a>.

%H Zhi-Wei Sun, <a href="https://doi.org/10.1142/S1793042119501045">Restricted sums of four squares</a>, Int. J. Number Theory 15(2019), 1863-1893.  See also <a href="http://arxiv.org/abs/1701.05868">arXiv:1701.05868 [math.NT]</a>.

%H Zhi-Wei Sun, <a href="https://arxiv.org/abs/2010.05775">Sums of four squares with certain restrictions</a>, arXiv:2010.05775 [math.NT], 2020.

%e a(1) = 1, and 2*1 + 1 = 1^2 + 0^2 + 1^2 + 1^2 with 1 + 2*0 + 3*1 = 2^2.

%e a(3) = 1, and 2*3 + 1 = 1^2 + 2^2 + 1^2 + 1^2 with 1 + 2*2 + 3*1 = 2^3.

%e a(9) = 1, and 2*9 + 1 = 1^2 + 6^2 + 1^2 + 1^2 with 1 + 2*6 + 3*1 = 2^4.

%e a(21) = 1, and 2*21 + 1 = 5^2 + 4^2 + 1^2 + 1^2 with 5 + 2*4 + 3*1 = 2^4.

%e a(39) = 1, and 2*39 + 1 = 1^2 + 5^2 + 7^2 + 2^2 with 1 + 2*5 + 3*7 = 2^5.

%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];

%t PQ[n_]:=PQ[n]=n>1&&IntegerQ[Log[2,n]];

%t 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]

%Y Cf. A000079, A000118, A000290, A000302, A279612, A299924, A338094, A338095, A338103, A338119, A338121.

%K nonn

%O 0,3

%A _Zhi-Wei Sun_, Oct 09 2020