Remarkable Features of the Number System (4)

I next studied the factor composition for those natural numbers with repeating prime structures.

For example from my prime factor generator at the 15 digit level,

900000000000414 = 22441 * 7549 * 4157 * 71 * 3 * 3 * 2.

So this represents a natural number with repeating prime structures that contains in all 7 prime factors. (1)

However if we were to count factors (where each distinct prime is counted only once) we would then have 6 prime factors ( as 3 recurs twice). (2)

Finally if we were to count factors where any recurring factor is excluded from consideration, we would obtain just 5 prime factors. (3)

So I sought to find with respect to the natural number system as a whole, the relationship between (1) and (2), then the relationship between (2) and (3) and finally the relationship as between (1) and (3).

Though still at a comparatively early stage of the number system - where numbers typically are still composed of relatively few prime factors - I could see a distinctive pattern beginning to emerge, which I will now outline for further empirical and theoretical investigation.

Therefore with respect to the combined factors of all natural numbers with repeating prime structures, the ratio of (1) where all the prime factors are included to (2) where any recurring prime is counted just once, ~ √2 (when n is sufficiently large).

The corresponding ratio of (2) where again each recurring prime is counted just once to (3). where recurring primes are excluded entirely from consideration, also  ~ √2 (when n is sufficiently large).


This therefore directly implies that the corresponding ratio of (1), where once more all prime factors are included to (3) where recurring prime factors are completely excluded   ~  2 (when n is sufficiently large).

Another way of expressing this result is that with respect to the average number of prime factors (for each natural number with a repeating prime structure) that the ratio of (1) to (2) ~ √2, that the ratio of (2) to (3) also ~ √2 and that finally the ratio of (1) to (3) ~  2 (when n is sufficiently large).

Expressed in yet another alternative manner, this would imply again - with respect to the natural numbers with repeating prime structures - that the average number of non-repeating prime factors would approximately equal the corresponding average number of recurring prime factors. And this approximation would become ever more accurate as n increases.

And because the overall number of factors for natural numbers with repeating and non-repeating prime structures respectively approaches equality (as n increases), this would therefore imply that the non-recurring prime factors (with respect to the natural numbers with repeating prime structures) would approximate 1/2 the total for corresponding factors of natural numbers (with non-repeating prime structures).