Eureka sequences

The two Eureka sequences first appeared in an issue of Eureka magazine, a journal published by the mathematical society of Cambridge University.
These two sequences, although simple, are quite puzzling. Once you figure out the answers though, you’ll realize how clear they were in the beginning.

Here is Eureka sequence number one:
3, 4, 6, 8, 12, …

The nature of these sequences is to reveal only a few terms, making this an even more vague and perplexing question. What do you think comes next?

You may try to find a correlation between the differences of each consecutive term. For example, 3 plus 1 is 4, and adding 2 to 4 gives 6, and so on. However, you will soon realize that there is no apparent pattern so far. Take a moment to think about what the next term in the sequence could be, and try to find the pattern rather than just guessing.

The next term is actually 14.
The pattern is simply to add 1 to each number in the sequence of primes.

The sequence of primes is: 2, 3, 5, 7, 11, 13, …

Here is the other Eureka sequence: 5, 8, 12, 18, 24, …

What is the next term? See if you can figure out this pattern.

Hint: Use the sequence of primes.