OFFSET
1,5
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, 61).
We have verified this for n up to 5*10^6. See also A335624 for a similar conjecture.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..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.
EXAMPLE
a(21) = 1, and 21 = 2^2 + 1^2 + 0^2 + 4^2 with 3*2 + 10*1 + 36*0 = 4^2.
a(98) = 1, and 98 = 6^2 + 7^2 + 3^2 + 2^2 with 3*6 + 10*7 + 36*3 = 14^2.
a(203) = 1, and 203 = 5^2 + 3^2 + 5^2 + 12^2 with 3*5 + 10*3 + 36*5 = 15^2.
a(760) = 1, and 760 = 0^2 + 18^2 + 20^2 + 6^2 with 3*0 + 10*18 + 36*20 = 30^2.
MATHEMATICA
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
TQ[n_]:=TQ[n]=n>0&&SQ[n];
tab={}; Do[r=0; Do[If[SQ[n-x^2-y^2-z^2]&&TQ[3x+10y+36z], 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, 1, 100}]; Print[tab]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Oct 08 2020
STATUS
approved