[teytra]

When I was younger I had a period I often was thinking about prime numbers (before I got old and started thinking about the Roman Empire).

I noticed the same as you, and IIRC the (some?) ancient greeks actually had an idea about 1 as not a number, but the unit that numbers were made of. So in a different class.

2 and 3 are also different, or rather all other primes from 5 and up are neighbours to a multiple of 6, (though not all such neighbours are primes of course).

In base-6 all those primes end in 5 or 1. What is the significance? I don't know. I remember that I started thinking that 2*3=6, maybe the sequence of primes is a result of the intertwining of numbersystems in multiple dimensions or whatever? Then I started thinking about the late republic instead. ;)

[alganet]

If you work not only the primes, but also the modulus function value of each non-prime, things get even more interesting than thinking of base changes! To me, it reveals much more.

[alganet]

Also, rearrangements.

In two dimensions is easier.

I cannot rearrange one pebble.

I can rearrange two or three pebbles equidistant from each other in just one distinct way (inverting the position of a neighbouring pebble).

And so on...

There are many ways to think of natural numbers without actual numbers.