Number and sum of prime factors of a number

[suchith]

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

 

[pbuk]

If such a formula $f(N)$ exists, what would $f(N) = 1$ mean?

 

[uperkurk]

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.

 

[pbuk]

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.

 

[CellCoree]

.

I can say that there is no known formula to directly find the number and sum of prime factors of a given large number N without listing or factorizing them. The functions τ(N) and σ(N) are commonly used to calculate the number and sum of all factors of a given number, not just the prime ones. These functions require the prime factorization of N, which involves listing or factorizing.

There may be other methods or algorithms that can be used to indirectly find the number and sum of prime factors without listing or factorizing them, but these have not been discovered or widely studied yet. The best approach currently would be to list or factorize the prime factors of N and then use the τ(N) and σ(N) functions to calculate the desired values. This may be time-consuming for very large numbers, but it is the most accurate and reliable method available.

In conclusion, while there is no known formula to directly find the number and sum of prime factors without listing or factorizing, scientists are continuously researching and exploring new methods to solve this problem.