I've been wondering, which in addition to twelve being so awesome is why I signed up here, if, for whatever reason, it was somehow necessary to use an odd number base which would be "best". Or rather, I suppose, which bases would be most useful, and why?

I've tried looking around on the 'net a bit, but I haven't managed to really find anything, so I thought I'd ask here. Thanks in advance.

All we mean when we say "odd" is that an integer is indivisible by the commonest and simplest prime number, 2. If we are after a number base which is highly divisible, we can certainly find them among odd numbers.

If we consider "prime factorization formulas" for some even bases, we can find solutions that avoid the prime 2. These solutions will be just as versatile in terms of divisibility as their even cousins.

Bases, 6, 10, 14 represent the "prime shape" x*y. Their factorizations are 2*3, 2*5, and 2*7, respectively. The numbers 15, 21, and 35 employ the same "shape", x*y, in their factorizations 3*5, 3*7, and 5*7. We can think of bases with the same "prime shape" or "prime factorization formulas" as "cousins". So bases 6, 10, 14, 15, 21, 35, etc are "cousins" with the same number of divisors, 6 being the simplest manifestation of the "prime shape", the "eldest cousin".

Base 12, with the "prime shape" of x^2 * y, has a cousin in the numbers 45 (3^2 * 5), 63 (3^2 * 7), 75 (3 * 5^2), 175 (5^2 * 7), etc. These numbers feature the same degree of divisibility (with different divisors); they'll enjoy the same number of divisors, and thus the same versatility. Base 12 is the simplest mainfestation of x^2 * y (note that x and y are dissimilar).

A key drawback of the odd base is evident when you compare the scales of the number bases. Skipping the simplest prime, 2, comes at the price of inflating the base. So the odd composite bases are always larger than the even bases of the same "prime shape". The first odd squarefree biprime (shape x*y) is 15, while the simplest even squarefree biprime is 6. This also has the effect of injecting plenty of "totative" factors (numbers t smaller than the base r which have a least common multiple equal to t*r.) Twelve's totatives are 1, 5, 7, and 11. Forty-five has far more totatives: 24 of them; these have the same "shape" x^2 * y. In fact, 45 is so inefficient that you can "buy" divisibility by five at base 60 for fewer totatives (only 16).

Another important class of odd bases are prime bases. These are poor in divisibility and rich in totatives. (The totient value of any prime base r is r-1: so base 11 has 10 totatives, base 23 has 22 totatives. All digits below the base "don't come out" cleanly as fractions, their multiplication facts do not feature patterns that facilitate memorization because their periods of repetition are r). Odd prime bases (meaning any prime but 2) do have their uses in cryptography and security. If you are in need of divisibility and ease of computation in a base, they aren't as valuable.

Hmm... if we were a dozenal civilization, would, should we have a word for "divisible by three"? Triadic? Would that be more plausible than having a term for "divisible by five" in our decimal world? "Fivable?"

Relatedly, should we have a word for "divisible by four"? Since that's equivalent to being divisible by two twice, I sometimes refer to "even even" numbers. As in, the Summer Olympics are held in even even years and the Winter Olympics are held in odd even years.

I like eveneven and oddeven as you've used them. I like "threeable" and "fivable", "sevenable", etc. but they sound naive, something my kindergartener would say. Keiridai is keen in calling out how society places some special consideration to even numbers as the rationale isn't perhaps evident as it ought to be.

So what about every eight years, would those be eveneveneven? Would it be "eightable"? Would every sixth be "sixable"?