# Estimates for Sum of Prime Factors and Number of Prime Factors

Given a positive integer $n$, I've workout out a formula which involves the expression "sum of distinct primes dividing n" minus "number of distinct prime factors of n."

Are there any known inequalities for "sum of distinct primes dividing n" or for "number of distinct prime factors of n" that hold for all $n$?

I'm not interested in asymptotics.

1. number of distinct prime factors: http://oeis.org/A001221
2. sum of distinct prime factors: http://oeis.org/A008472
• What you ask in the question is not quite complete and understandable !! Please try to edit your question and make it more precise. You are just giving a function (or rather a number) and asking to find inequalities about that !! You see there are plenty of inequalities that can be created with a given number and that too without any computer aid and if you use a computer then you will get mesmerizing results and lots and lots of inequalities. So you edit this question or do something to make it more precise and exact. – adityaguharoy Jul 22 '17 at 6:07

There are a number of obvious inequalities (thus "known" in the sense of "derivable by elementary methods") but they are usually too weak to be of interest. For prime numbers n, the sum of primes dividing them is n, for numbers of the form $p^aq^b$ the sum of their distinct prime factors is $p+q$, which is usually less than n and is less than n/2 when p is greater than 2 and q is greater than 5, but it is not clear how one goes from there or what one does with the inequalities derived. Similarly, the number of distinct prime factors of n is (for n not too small) less than log n , and is almost always less than (log n)/(log (log n)) (maybe there are 5 exceptions to this?), and again there are not so many uses for this. However, I am using these latter estimates in analysis of the resource usage of some number theory programs I am writing. Perhaps you could think of applications for such formulas and then search for those applications.
It is known that $$\omega(n):=\sum_{p\mid n}1\leq(1 + o(1))\frac{\log n}{\log\log n},$$ and this is optimal (see e.g. Section 5.3 in Tenenbaum: Introduction to analytic and probabilistic number theory). Regarding the second quantity, we have the following lower bound that is optimal under $\omega(n)\to\infty$: $$\sum_{p\mid n} p\geq \left(\frac{1}{2}+o(1)\right)\omega(n)^2\log\omega(n).$$ See also this somewhat related MO question.