I have a series of 30 given numbers as given : $0, 2, 4, 5, 6, 7, 7, 9, 8, 8, 9, 13, 9, 15, 11, 10, 10, 19, 10, 21, 11, 12, 15, 25, 11, 12, 17, 11, 13, 31$ and $12$.
This was generated by the following method :
First I took all even numbers upto 60, starting from 0.
Then, I took out all its prime factors. Prime factors that were repeated were treated as explicit factors.
Then I added all prime factors (of each even number) up (including repeating ones) and wrote them alongside.
This formed a series of numbers as given in the top. This (pattern?) should somehow depend on the pattern of prime numbers. Is it possible to write a mathematical function for generating the nth number of this series ?
If you don't restrict to even numbers, you get https://oeis.org/A001414 and you will find there are many references and links at that site. No "mathematical function" is given (although that depends on what exactly you mean by that phrase). If the prime factorization of $n$ is $n=p_1^{a_1}p_2^{a_2}\dotsb p_r^{a_r}$, then your function is $\sum_{j=1}^ra_jp_j$