The orders of the simple groups (ignoring the matrix groups for which the problem is solved) all seem to be a lot like this:
2^46 3^20 5^9 7^6 11^2 13^3 17 19 23 29 31 41 47 59 71
starts with a very high power of 2, then the powers decrease and you get a tail - it's something like exponential decay.
Why does this happen? I want to understand this phenomenon better.
I wanted to find counter-examples, e.g. a simple group of order something like
2^4 3^2 11^5 13^9
but it seems like they do not exist (unless it slipped past me!).
We have the following bound $|G| \le \left(\frac{|G|}{p^k}\right)!$ which allows $3^2 11^4$ but rules out orders like $3^2 11^5$, $3^2 11^6$, .. while this does give a finite bound it is extremely weak when you have more than two primes, it really doesn't explain the pattern but a much stronger bound of the same type might?
I also considered that it might be related to multiple transitivity, a group that is $t$-transitive has to have order a multiple of $t!$, and e.g. 20! =
2^18 3^8 5^4 7^2 11 13 17 19
which has exactly the same pattern, for reasons we do understand. But are these groups really transitive enough to explain the pattern?
I am posting the following counterexample to the question, as requested by caveman in the comments.
The Steinberg group ${}^2A_5(79^2)$ has order $$ 2^{23}\cdot 3^4\cdot 5^6\cdot 7^2\cdot 11^1\cdot 13^3\cdot 43^1\cdot 79^{15}\cdot 641^1\cdot 1091^1\cdot 3121^1\cdot 6163^2.$$
There are other counterexamples, too. For example ${}^2A_9(47^2)$ has order $$ 2^{43}\cdot 3^{13}\cdot 5^2\cdot 7^3\cdot 11^1\cdot 13^2\cdot 17^2\cdot 23^5\cdot 31^1\cdot 37^1\cdot 47^{45}\cdot 61^1\cdot 97^1\cdot 103^3\cdot 3691^1\cdot 5881^1\cdot 14621^1\cdot 25153^1\cdot 973459^1\cdot 1794703^1\cdot 4778021^2.$$
I would guess there are infinite counterexamples, but the numbers (of course) get very very large!