OFFSET
1,3
COMMENTS
Conjecture: a(n) exists for any n > 0.
See also A281976 for a related conjecture.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..300
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017.
EXAMPLE
a(1) = 0 since 0 = 0^2 + 0^2 + 0^2 + 0^2 with 0 = 0^2 and 0 + 24*0 = 0^2.
a(2) = 1 since 1 = 0^2 + 0^2 + 0^2 + 1^2 with 0 = 0^2 and 0 + 24*0 = 0^2, and 1 = 1^2 + 0^2 + 0^2 + 0^2 with 1 = 1^2 and 1 + 24*0 = 1^2.
a(4) = 10 since 10 = 0^2 + 0^2 + 1^2 + 3^2 with 0 = 0^2 and 0 + 24*0 = 0^2, 10 = 1^2 + 0^2 + 0^2 + 3^2 with 1 = 1^2 and 1 + 24*0 = 1^2, 10 = 1^2 + 1^2 + 2^2 + 2^2 with 1 = 1^2 and 1 + 24*1 = 5^2, and 10 = 1^2 + 2^2 + 1^2 + 2^2 with 1 = 1^2 and 1 + 24*2 = 7^2.
MATHEMATICA
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Do[m=0; Label[aa]; r=0; Do[If[SQ[m-x^4-y^2-z^2]&&SQ[x^2+24y], r=r+1; If[r>n, m=m+1; Goto[aa]]], {x, 0, m^(1/4)}, {y, 0, Sqrt[m-x^4]}, {z, 0, Sqrt[(m-x^4-y^2)/2]}]; If[r<n, m=m+1; Goto[aa], Print[n, " ", m]]; Continue, {n, 1, 60}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Feb 09 2017
STATUS
approved