A373997 Greatest positive integer k for which the y-coordinates of the extreme points and the inflection point of y = f(x) = 1/k*(x - A373995(n))*(x - A373996(n)) are integers.

2, 2, 16, 2, 2, 2, 16, 54, 2, 54, 128, 16, 2, 16, 2, 16, 128, 250, 2, 2, 250, 432, 54, 54, 2, 2, 2, 16, 686, 54, 432, 2, 1024, 128, 686, 16, 128, 16, 2, 2, 1458, 128, 1024, 2, 2, 2000, 250, 54, 250, 1458, 2, 16, 2662, 2, 16, 2, 250, 2000, 2, 3456, 432, 54, 432

Offset: 1

Author

Felix Huber, Jul 07 2024

Keywords

• nonn

Examples

			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.
		

Crossrefs

Cf. A373995 (values x1), A373996 (values x2), A364384, A364385.

Programs

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);

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)).

Extensions

Data corrected by Felix Huber, Aug 18 2024