What is the maximum number of positive rational values that can be obtained by combining $k$ positive integers using only addition, subtraction, multiplication, division, and parentheses? Assume that each value must be used once and only once.
I believe that the sequence starts as follows: 1, 5, 47, 733, 15907, 443825.
The second term is 5 because we can form 5 different positive numbers from $\{a, b\}$ if $a > b$:
$a+b,\ \ a-b,\ \ a\cdot b,\ \ \frac{a}b,\ \ \frac{b}a\ .$