## Number and sum of prime factors of a number

Given a large number N, do we have any formula to find the number of prime factors and their sum like τ(N) and σ(N) functions?

CONDITION: One should not list the factors of N or is not allowed to factorize N since afterwards it would be just a matter of counting and addition
 If such a formula ## f(N) ## exists, what would ## f(N) = 1 ## mean?
 I believe if such a formula does exist then the entire internet would be vulnerable. The internet is secure because of prime number factorisation. See RSA Algorithm There is a formula to find if a number is prime or not, but not the factors. $\Large{{p_n=1+\displaystyle\sum_{k=1}^{2 \cdot (\lfloor n\ln(n) \rfloor+1)}\left(1-\left\lfloor \frac{1}{n} \cdot \displaystyle\sum_{j=2}^k \left\lceil \frac{3-\displaystyle\sum_{i=1}^j \left\lfloor \frac{\left\lfloor \frac{j}{i} \right\rfloor}{\left\lceil \frac{j}{i} \right\rceil} \right\rfloor}{j} \right\rceil \right\rfloor\right)}$

## Number and sum of prime factors of a number

That actually looks like a "formula" to find the ## n ##th prime, but such "formulas" are really just symbolic descriptions of (very ineffecient) algorithms and of no practical importance.