Disclaimer : I will try to be as neutral as possible and let the numbers speak, but interpretations will inevitably comes with the risk of mistakes. But you'll have the tools (you always had) to control each results.

If numbers bores you, wait a little bit and come back soon...

This first post is about the main Tools & nature of numbers.

I) Function F (as in function) and considerations about primes.

I call F the function which associate to any integer number the sum of it primes factors & the sum of their ranks in the list of prime numbers (I call if needed Ff the prime part of F & Fr the rank part).

If you think there is something artificial with this 'construct' you're just can't be more wrong about anything. You'll never find in any Universe something more natural than the natural numbers (precisely). And inside you'll find all numbers are composite of a few of them : the Primes, in which the Order is like a Holy Quest Grail, in fact nearly all mathematicians think this order doesn't exist. So, to me, primes are the archetypes of all things and also their ranks in this list.

examples :

30 = 2*3*5 (of course prime factorisation is unique) = 1st prime * 2nd prime * 3rd prime
So F(30) = Ff(30) + Fr(30) = 2+3+5 + 1+2+3 = 10 + 6 = 16

7 = 7 (yes, I'm pretty sure) = 4th prime
-> F(7) = 11 [7 is associated to 4 by it's ranks, I'm sure most of you know some use the 7_4 for GOD. 7 is also associated closely to 11 [maybe you read a thing or two about this number since those last eleven years...].

F can be itered without limit, ex :
F(1000000)=66, F(66)=24, F(24)=14.
One could write, F3(1000000) = 14

It happens there is only 3 endings possible to Fn :

The last value can be 14 (F(14)=14=2*7)

The value follow a 3-period cycle
F(11)=16, F(16)=12, F(12)=11 (sum 39=3*13)

The value follow a 4-period cycle
F(19)=27, F(27)=15, F(15)=13, F(13)=19 (sum 74=2*37)

And there is, as always, the One case : F(1)=1

Total sum = 128 = 2^7

Which leads us to the 1st intermission.