Answer:

31

Step-by-step explanation:

The answer to this question is 31.

We can prove this by showing that every integer greater than 31 can be expressed as the sum of distinct prime numbers raised to distinct prime powers.

First, we observe that any even integer greater than 2 can be expressed as the sum of two distinct prime numbers (by the Goldbach Conjecture, which has been verified for all numbers up to at least $10^{18}$). Thus, any even integer greater than 2 raised to a prime power can be expressed as the sum of two distinct prime numbers raised to distinct prime powers.

Next, we consider odd integers greater than 31. By Bertrand's Postulate, there is always a prime between $n$ and $2n$ for any integer $n\geq2$. Thus, any odd integer greater than 31 can be expressed as the sum of two distinct odd prime numbers. If one of these primes is 2, then the sum can be expressed as the sum of three distinct prime numbers raised to distinct prime powers (since 2 to any positive integer power is still 2). Otherwise, both primes are odd and can be raised to the first power, giving a sum of two distinct prime numbers raised to distinct prime powers.

Therefore, the only positive integer that cannot be expressed as the sum of distinct prime numbers raised to distinct prime powers is 31.