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A272888
Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with w*(x^2 + 8*y^2 - z^2) a square, where w,x,y are nonnegative integers and z is a positive integer.
19
1, 2, 2, 1, 4, 5, 1, 2, 5, 5, 4, 4, 5, 8, 2, 2, 8, 6, 4, 6, 9, 5, 3, 4, 5, 12, 9, 1, 11, 8, 4, 2, 8, 9, 8, 7, 6, 12, 1, 5, 14, 10, 4, 8, 15, 9, 3, 4, 8, 14, 11, 5, 11, 16, 2, 6, 11, 6, 11, 4, 13, 13, 1, 1, 16, 17, 6, 9, 13, 9, 5, 7, 9, 19, 12, 6, 17, 8, 4, 6
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, 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}]
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