IR-TST 2018 Problem 10

Call a positive integer "useful but not optimized " (!), if it can be written as a sum of distinct powers of $3$ and powers of $5$ . Prove that there exist infinitely many positive integers which they are not "useful but not optimized". (e.g. $37=(3^0+3^1+3^3)+(5^0+5^1)$ is a " useful but not optimized" number)

Proposed by Mohsen Jamali