Remarkable Features of the Number System (2)

We have seen in the last entry that the average gap as between natural numbers with non-repeating prime structures ~ 1 + 2/π.

The corresponding average gap as between natural numbers with repeating prime structures
~ 1 + π/2.

This therefore entails that the frequency of the natural numbers up to n with non-repeating prime structures ~ n/(1 + 2/π).

The corresponding frequency of the natural numbers up to n with repeating prime structures ~ n/(1 + π/2).

I next looked at the combined number of factors for numbers with repeating structures as opposed to those with non-repeating prime structures.

In the earlier stages of the number system, the combined frequency of factors for natural numbers with repeating prime structures predominates over those with non-repeating structures. However as one ascends the number scale, this imbalance starts to steadily fall. So whereas initially - say from 1000 to 2000 - the ratio of combined factors of numbers with non-repeating to numbers with repeating prime structures is about .75, higher up the number system (in the vicinity of 10¹⁵) the corresponding ratio is close to .9.

Now this might seem like a slow increase relative to the size of natural numbers involved. However relative to the combined number of factors it is in fact very rapid. For it must be remembered that the average number of distinct prime factors per (natural) number rises very slowly (i.e. at the rate of log log n)!

So the assumption here is that with a sufficient increase in the average frequency of prime factors (for each number) that the ratio of the combined frequency of factors - for natural numbers with non-repeating prime structures - to that with repeating structures approaches 1.

This would then further imply that the  ratio of the average number of factors - for each individual natural number with repeating prime structures - to that with non-repeating structures ~ 1 + 2/π.