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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.

%I A373997 #15 Aug 19 2024 11:41:09
%S A373997 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,
%T A373997 2,2,16,686,54,432,2,1024,128,686,16,128,16,2,2,1458,128,1024,2,2,
%U A373997 2000,250,54,250,1458,2,16,2662,2,16,2,250,2000,2,3456,432,54,432
%N 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.
%F A373997 x-coordinate of the 1. extreme point: x3 = (x1 + x2 + sqrt(x1^2 + x2^2 - x1*x2))/3.
%F A373997 x-coordinate of the 2. extreme point: x4 = (x1 + x2 - sqrt(x1^2 + x2^2 - x1*x2))/3.
%F A373997 x-coordinate of the inflection point: x5 = (x1 + x2)/3 = (x3 + x4)/2.
%F A373997 k = GCD(f(x3), f(x4), f(x5)).
%e A373997 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.
%p A373997 A373997:=proc(s)
%p A373997   local x_1,x_2,x_3,x_4,x_5,L;
%p A373997   L:=[];
%p A373997   for x_1 from 1 to floor((s-1)/2) do
%p A373997     x_2:=s-x_1;
%p A373997     x_3:=(x_1+x_2+sqrt(x_1^2+x_2^2-x_1*x_2))/3;
%p A373997     x_4:=(x_1+x_2-sqrt(x_1^2+x_2^2-x_1*x_2))/3;
%p A373997     if x_3=floor(x_3) and x_4=floor(x_4) then
%p A373997       x_5:=(x_3+x_4)/2;
%p A373997       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))];
%p A373997     fi;
%p A373997   od;
%p A373997   return op(L);
%p A373997 end proc;
%p A373997 seq(A373997(s),s=3..414);
%Y A373997 Cf. A373995 (values x1), A373996 (values x2), A364384, A364385.
%K A373997 nonn
%O A373997 1,1
%A A373997 _Felix Huber_, Jul 07 2024
%E A373997 Data corrected by _Felix Huber_, Aug 18 2024