Except for three, no primes in nonary end in 0, 3 or 6, since any nonary number ending in 0, 3 or 6 is divisible by three.
A nonary number is divisible by two, four or eight, if the sum of its digits is also divisible by two, four or eight respectively.
If x is a triangular number, so is 9x+1. This means that one finds 3, 31, 311, 3111, 31111... in the triangular numbers. Likewise, 6, 61, 611, 6111, ....
Although the term "Nonary" is used in describing the written form of the language used by the fictional civilization, The Culture, found in Iain M. Banks' books, the description on page 119 of Excession reads more like it's based on a binary system with a 9-bit 'byte'.