Thursday, March 10, 2016

Average Frequency of All Prime Factors of a Number.

As is well known the Hardy-Ramanujan Theorem relates to the average frequency of (distinct) prime factors of a number and ~ log log n.

So once again if we take a simple number such as 28 to illustrate its unique prime factor expression is is given as 2 * 2 * 7. This implies that it contains 3 prime factors in all. However it contains only 2 distinct prime factors (i.e. 2 and 7).

So the question remains as to what expression gives the average frequency of all prime factors of a number!

Now in my previous blog entries here, from investigation of the prime factors of numbers up to
1016, I reached the tentative conclusion that the combined number of factors for both natural numbers with repeating prime and non-repeating prime factor structures eventually approaches equality with each other.

Initially, the combined total of factors for natural numbers with repeating prime factor structures is definitely greater than the corresponding total for numbers with non-repeating prime factor structures. How this relative dominance tends to fall, so that when n = 1016, the total (for non-repeating primes) has reached about 90% of the corresponding total (for repeating primes).

And my hypothesis is that this would gradually approach 100% (as we ascended higher on the number scale). We must remember that even though 1016 already seems to represent an advanced point on the number scale, the average number of factors for each number is still very low. So the average frequency of factors itself needs to significantly increase, before the true pattern of behaviour in this context becomes properly apparent!

Also, I had found tentative evidence for those numbers with repeating prime factor combinations that the ratio of all factors to (distinct) prime factors ~ √2 (with again this approximation steadily improving as we ascend the number scale).

Therefore combining both results, this would suggest that the average frequency of all prime factors would ~ (1 + √2)/2{log log n}. 

In other words, for sufficiently high values of n, we would expect on average roughly 20% more prime (i.e. all prime) than (distinct) prime factors for each number.

Of course, over the earlier stretches of the number system, we would expect a much higher difference than 20%.

For example up to n = 100, the relative frequency  of (all) prime to that of (distinct) prime factors is  about 40%. However this percentage  would continue to fall as we ascend higher on the number scale until it eventually ~ (1 + √2)/2.