What's the next number in this sequence?

Find the next number in this sequence:

2, 4, 6, 9, 14, 18, 25, 32, 44, ?

Bonus question: I originally was going to use a different related sequence which grew at a faster rate, but changed it to this one instead. What was the original sequence, and why might I have opted for this one instead?

The next number is

63

The sequence seems to be

$a_n = F_n + P_n$, the $n$-th Fibonacci number plus the $n$-th prime number

0+2, 1+3, 1+5, 2+7, 3+11, 5+13, 8+17, 13+19, 21+23, so the next number is 34+29

And my guess for bonus,

perhaps you originally multiplied, but didn't like that because it'd have been more obvious if we'd factored the numbers

• Correct on all accounts. That was exactly my reasoning for the bonus question. – TTT Feb 1 '16 at 22:15