OFFSET
1,2
COMMENTS
Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 7, 39, 63, 87, 5116, 2^(4k+2)*m (k = 0,1,2,... and m = 1, 7).
See arXiv:1604.06723 for more refinements of Lagrange's four-square theorem.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, arXiv:1604.06723 [math.GM], 2016.
Zhi-Wei Sun, Refine Lagrange's four-square theorem, a message to Number Theory List, April 26, 2016.
EXAMPLE
a(1) = 1 since 1 = 0^2 + 0^2 + 0^2 + 1^2 with 1 > 0 and 0*(0^2 + 8*0^2 - 1^2) = 0^2.
a(4) = 1 since 4 = 0^2 + 0^2 + 0^2 + 2^2 with 2 > 0 and 0*(0^2 + 8*0^2 - 2^2) = 0^2.
a(7) = 1 since 7 = 2^2 + 1^2 + 1^2 + 1^2 with 1 > 0 and 2*(1^2 + 8*1^2 - 1^2) = 4^2.
a(28) = 1 since 28 = 2^2 + 2^2 + 4^2 + 2^2 with 2 > 0 and 2*(2^2 + 8*4^2 - 2^2) = 16^2.
a(39) = 1 since 39 = 1^2 + 3^2 + 2^2 + 5^2 with 5 > 0 and 1*(3^2 + 8*2^2 - 5^2) = 4^2.
a(63) = 1 since 63 = 2^2 + 5^2 + 3^2 + 5^2 with 5 > 0 and 2*(5^2 + 8*3^2 - 5^2) = 12^2.
a(87) = 1 since 87 = 2^2 + 1^2 + 9^2 + 1^2 with 1 > 0 and 2*(1^2 + 8*9^2 - 1^2) = 36^2.
a(5116) = 1 since 5116 = 65^2 + 9^2 + 9^2 + 27^2 with 27 > 0 and 65*(9^2 + 8*9^2 - 27^2) = 0^2.
MAPLE
N:= 1000; # to get a(1)..a(N)
A:= Vector(N):
for z from 1 to floor(sqrt(N)) do
for x from 0 to floor(sqrt(N-z^2)) do
for y from 0 to floor(sqrt(N-z^2-x^2)) do
q:= x^2 + 8*y^2 - z^2;
if q < 0 then
A[x^2+y^2+z^2]:= A[x^2+y^2+z^2]+1
elif q = 0 then
for w from 0 to floor(sqrt(N-z^2-x^2-y^2)) do
m:= w^2 + x^2 + y^2 + z^2;
A[m]:= A[m]+1;
od
else
wm:= mul(`if`(t[2]::odd, t[1], 1), t=isqrfree(q)[2]);
for j from 0 to floor((N-z^2-x^2-y^2)^(1/4)/sqrt(wm)) do
m:= (wm*j^2)^2 + x^2 + y^2 + z^2;
A[m]:= A[m]+1;
od;
fi
od
od
od:
convert(A, list); # Robert Israel, May 27 2016
MATHEMATICA
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0; Do[If[SQ[n-x^2-y^2-z^2]&&SQ[Sqrt[n-x^2-y^2-z^2](x^2+8y^2-z^2)], r=r+1], {x, 0, Sqrt[n-1]}, {y, 0, Sqrt[n-1-x^2]}, {z, 1, Sqrt[n-x^2-y^2]}]; Print[n, " ", r]; Continue, {n, 1, 80}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, May 08 2016
EXTENSIONS
Rick L. Shepherd, May 27 2016: I checked all the statements in each example.
STATUS
approved