OFFSET
0,2
COMMENTS
Conjecture: a(n) > 0 if n is not divisible by 8. Moreover, a(n) = 0 if and only if n has the form 2^(4k+3)*m (k >= 0 and m = 1, 3, 5, 43).
We have verified this for n up to 3*10^6. The conjecture is similar to the author's 1-3-5 conjecture (cf. A271518).
In his 2017 JNT paper, the author conjectured that any natural number not of the form 2^(4k+2)*7 (k = 0,1,...) can be written as w^2 + x^2 + y^2 + z^2 with w + 2*x + 3*y + 5*z a square, where w, x, y, z are nonnegative integers.
See also A338019 for a similar conjecture.
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, Sums of four squares with certain restrictions, arXiv:2010.05775 [math.NT], 2020.
MAPLE
a(4) = 1, and 4 = 0^2 + 0^2 + 0^2 + 2^2 with 0 + 3*0 + 4*0 = 0^2.
a(7) = 1, and 7 = 2^2 + 1^2 + 1^2 + 1^2 with 2 + 3*1 + 4*1 = 3^2.
a(44) = 1, and 44 = 3^2 + 3^2 + 1^2 + 5^2 with 3 + 3*3 + 4*1 = 4^2.
a(328) = 1, and 328 = 8^2 + 16^2 + 2^2 + 2^2 with 8 + 3*16 + 4*2 = 8^2.
a(776) = 1, and 776 = 24^2 + 0^2 + 10^2 + 10^2 with 24 + 3*0 + 4*10 = 8^2.
MATHEMATICA
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={}; Do[r=0; Do[If[SQ[n-x^2-y^2-z^2]&&SQ[x+3y+4z], r=r+1], {x, 0, Sqrt[n]}, {y, 0, Sqrt[n-x^2]}, {z, 0, Sqrt[n-x^2-y^2]}];
tab=Append[tab, r], {n, 0, 100}]; Print[tab]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Oct 08 2020
STATUS
approved