Some numbers appear frequently, as you would expect (primes, numbers 1 off from prime, etc) while others appear infrequently, but in general there is a downward trend as the numbers become larger. That is all expected; what isn't expected is that there seems to be a break between common numbers and uncommon numbers. A very distinct break. The theory is that social prejudices towards certain bases and certain numbers have caused mathematicians to find more sequences with numbers that they "like" than those that they don't. So for example, numbers that look neat in base 10, say "3333" or "4444" get special treatment and tend to occur in more sequences.

The paper on this weird distribution is rather accessible, I recommend it: http://arxiv.org/abs/1101.4470 Page 4 has the pretty plot.