OFFSET
1,1
FORMULA
x-coordinate of the 1. extreme point: x3 = (x1 + x2 + sqrt(x1^2 + x2^2 - x1*x2))/3.
x-coordinate of the 2. extreme point: x4 = (x1 + x2 - sqrt(x1^2 + x2^2 - x1*x2))/3.
x-coordinate of the inflection point: x5 = (x1 + x2)/3 = (x3 + x4)/2.
k = GCD(f(x3), f(x4), f(x5)).
EXAMPLE
a(3) = 16, since y = f(x) = 1/16*(x - 18)*(x - 48) has the extrema (8, 200), (36, -486) and the inflection point (22, -143). Since GCD(200, -143, -486) = 1, there is no value of k > 16, for which the y-coordinates of these three points are all integers.
MAPLE
A373997:=proc(s)
local x_1, x_2, x_3, x_4, x_5, L;
L:=[];
for x_1 from 1 to floor((s-1)/2) do
x_2:=s-x_1;
x_3:=(x_1+x_2+sqrt(x_1^2+x_2^2-x_1*x_2))/3;
x_4:=(x_1+x_2-sqrt(x_1^2+x_2^2-x_1*x_2))/3;
if x_3=floor(x_3) and x_4=floor(x_4) then
x_5:=(x_3+x_4)/2;
L:=[op(L), gcd(gcd(x_3*(x_3-x_1)*(x_3-x_2), x_4*(x_4-x_1)*(x_4-x_2)), x_5*(x_5-x_1)*(x_5-x_2))];
fi;
od;
return op(L);
end proc;
seq(A373997(s), s=3..414);
CROSSREFS
KEYWORD
nonn
AUTHOR
Felix Huber, Jul 07 2024
EXTENSIONS
Data corrected by Felix Huber, Aug 18 2024
STATUS
approved