%I #17 Oct 21 2022 20:22:18

%S 15,20,24,30,35,40,42,45,48,55,56,60,63,65,66,70,72,75,77,78,80,84,85,

%T 88,90,91,96,99,100,104,105,110,112,117,119,120,126,130,132,135,136,

%U 140,143,144,150,152,153,154,156,160,161,165,168,170,171,175

%N Integers k such that there exist integers x,y,z from {1,2,...,k-1} for which x*y*z = (k-x)*(k-y)*(k-z) and the factors x, y, z are all different from any of k-x, k-y, k-z (this is always the case for odd k; and x,y,z must differ from k/2 for even k).

%H Robert Israel, <a href="/A102495/b102495.txt">Table of n, a(n) for n = 1..3500</a>

%p Filter:= proc(n) local x,y,z,q;

%p for x from 1 to n-1 do

%p for y from x to n-1 do

%p z:= (n-x)*(n-y)*n/(n^2-n*x-n*y+2*x*y);

%p if z::integer and not has({x,y,z},n/2) then return true fi;

%p od od;

%p false

%p end proc:

%p select(Filter, [$1..300]); # _Robert Israel_, Dec 30 2015

%t okQ[n_] := Do[z = (n-x)(n-y)n/(n^2 - n x - n y + 2 x y); If[IntegerQ[z] && AllTrue[{x, y, z}, FreeQ[#, (n-x)|(n-y)|(n-z)]&], Return[True]], {x, 1, n-1}, {y, x, n-1}];

%t Select[Range[200], okQ] (* _Jean-François Alcover_, Jun 16 2020 *)

%Y Cf. A102505, A102509.

%K nonn

%O 1,1

%A _Max Alekseyev_, Feb 26 2005