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

1,1

Links

Table of n, a(n) for n=1..63.

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

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

Sequence in context: A079897 A097540 A345225 * A112327 A152541 A302339

Adjacent sequences: A373994 A373995 A373996 * A373998 A373999 A374000

Keyword

nonn

Author

Felix Huber, Jul 07 2024

Extensions

Data corrected by Felix Huber, Aug 18 2024

Status

approved