What is the smallest positive integer that can not be written as a sum of positive integer powers of distinct elements of {1,…,n}?

Question

Lets say you have numbers from 1 to 3, how many numbers can be made using powers of the numbers without the base repeating?
$1=1^1$
$2=2^1$
$3=3^1$
$4=3^1+1^1$
$5=3^1+2^1$
$6=3^1+2^1+1^1$
$7=3^1+2^2$
$8=2^3$
$9=3^2$
$10=3^2+1^1$
$11=3^2+2^1$
$12=3^2+2^1+1^1$
$13=3^2+2^2$
$14=3^2+2^2+1^1$
The answer is 14 because 15 cannot be made without 2 base numbers repeating
What I'm asking is. Using the sum of powers with the base of 1 to n, without a base repeat. How many numbers can you make?