OFFSET
0,3
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.
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.
EXAMPLE
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.
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]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Oct 09 2020
STATUS
approved