OFFSET
0,2
COMMENTS
Conjecture: a(n) > 0 for all n >= 0. Moreover, if m > 1 has the form 2^a*(2*b+1), and either a is positive and even, or b is even, then m can be written as x^2 + y^2 + z^2 + w^2 with x^2 + 7*y^2 = 2^k for some positive integer k, where x, y, z, w are nonnegative integers.
We have verified the latter assertion in the conjecture for m up to 4*10^8.
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(0) = 1, and 4*0 + 1 = 1^2 + 0^2 + 0^2 +0^2 with 1^2 + 7*0^2 = 2^0.
a(25) = 2, and 25 = 2^2 + 2^2 + 1^2 + 4^2 = 4^2 + 0^2 + 0^2 + 3^2
with 2^2 + 7*2^2 = 2^5 and 4^2 + 7*0^2 = 2^4.
MATHEMATICA
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
PQ[n_]:=PQ[n]=IntegerQ[Log[2, n]];
tab={}; Do[r=0; Do[If[SQ[4n+1-x^2-y^2-z^2]&&PQ[x^2+7y^2], r=r+1], {x, 1, Sqrt[4n+1]}, {y, 0, Sqrt[4n+1-x^2]}, {z, 0, Sqrt[(4n+1-x^2-y^2)/2]}]; tab=Append[tab, r], {n, 0, 70}]; tab
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Oct 14 2020
STATUS
approved