A373995 -id:A373995 - cp's OEIS frontend

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

Felix Huber, Jul 07 2024

nonn

			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.

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

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);

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)).

Data corrected by Felix Huber, Aug 18 2024

A373996 Zeros x2 of polynomial functions f(x) = 1/kx(x - x1)*(x - x2), which have three integer zeros 0, x1 and x2 (with 0 < x1 < x2) as well as two extreme points and one inflection point with integer x-coordinates (sorted in ascending order, first by the sum x1 + x2 and then by x1).

Original entry on oeis.org

24, 24, 48, 45, 45, 63, 48, 72, 63, 72, 96, 90, 105, 90, 120, 126, 96, 120, 105, 144, 120, 144, 135, 135, 165, 120, 195, 126, 168, 189, 144, 144, 192, 180, 168, 210, 180, 240, 165, 240, 216, 252, 192, 288, 231, 240, 225, 189, 225, 216, 297, 210, 264, 195, 288, 231, 315, 240, 273, 288, 270, 315, 270

Offset: 1

Felix Huber, Jul 07 2024

nonn

The corresponding values x1 are in A373995. The corresponding maximum values for k, for which the y-coordinates of the extrema and the inflection are integers, are in A373997.
These polynomial functions can be used in math lessons when discussing curves. Zeros, extreme points and inflection points can be determined without unnecessary calculation effort with fractions and roots.
Of course, these functions can be stretched in the y-direction by a factor 1/k without affecting the zeros, the extreme points and the inflection point, or shifted in the x-direction, whereby the zeros, the extreme points and the inflection point are also shifted.

			24 is twice in the sequence, since the x-cordinates of the extreme points and of the inflection point of f(x) = 1/k*x*(x - 9)*(x - 24) are 4, 18 and 11 and of f(x) = 1/k*x*(x - 15)*(x - 24) are 6, 20 and 13.

Cf. A373995 (values x1), A373997 (maximum values for k), A364384, A364385.

A373996:=proc(s)
  local x_1,x_2,x_3,x_4,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
      L:=[op(L),x_2];
    fi;
  od;
  return op(L);
end proc;
seq(A373996(s),s=3..414);

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)).

Data corrected by Felix Huber, Aug 18 2024

cp's OEIS Frontend

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.

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A373996 Zeros x2 of polynomial functions f(x) = 1/kx(x - x1)*(x - x2), which have three integer zeros 0, x1 and x2 (with 0 < x1 < x2) as well as two extreme points and one inflection point with integer x-coordinates (sorted in ascending order, first by the sum x1 + x2 and then by x1).

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cp's OEIS Frontend

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.

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Author

Keywords

Examples

Crossrefs

Programs

Maple

Formula

Extensions

A373996 Zeros x2 of polynomial functions f(x) = 1/k*x*(x - x1)*(x - x2), which have three integer zeros 0, x1 and x2 (with 0 < x1 < x2) as well as two extreme points and one inflection point with integer x-coordinates (sorted in ascending order, first by the sum x1 + x2 and then by x1).

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Author

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Comments

Examples

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Programs

Maple

Formula

Extensions

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.

A373996 Zeros x2 of polynomial functions f(x) = 1/kx(x - x1)*(x - x2), which have three integer zeros 0, x1 and x2 (with 0 < x1 < x2) as well as two extreme points and one inflection point with integer x-coordinates (sorted in ascending order, first by the sum x1 + x2 and then by x1).