Usually, number theoretic functions have "trivial" (or at least easily defined) values for primes. In this thread, I am rather asking for **functions which are only defined on primes** (well, this condition is not absolutely essential, see below) **and which have an irregular behaviour**. But I’m only interested in functions such that $f(p)$ reflects some “intrinsic” property of $p$. This is to exclude functions like $f(p_n)=p_{n+1}-p_n$. Some examples that would ‘almost’ fall into this category:

- The smallest positive primitive root of $p_n$ (http://oeis.org/A001918) – but this is not only defined for primes.
- Functions linked to class numbers (like http://oeis.org/A126433, see also http://oeis.org/A002142) – but those are not specific to primes either.
- One could also think of the number of non isomorphic projective planes of order $p$, but I would not expect an "irregular behaviour" here, rather a strong monotonous growth. The functions I am asking for would rather
*not*have a combinatorical interpretation.

The example which has motivated this is the fact discovered here (look towards the end of the answer and in the comments) that for primes $p$, the structure of the modular curve $X_0(p^2)$ yields a sequence starting with $8,3,1,1,5,1$ for the first six primes. This (supposedly integer) sequence is not in the OEIS, and according to Michael Somos, finding more entries (i.e. for $p>13$) is probably beyond the limit of current computational power, as this implies searching for irreducible eta product identities with supposedly 100 or more terms and some coefficients of sizes larger than $10^6$. Unless some theoretical background is discovered, that is.

So I thought it might be a good idea to collect sequences of similar types – who knows if some of them happen to be linked to others? Maybe there are e.g. such sequences related to the Bernoulli numbers / zeta functions (but nothing "artificial" please). The underlying idea is of course to reveal more "hidden" information about primes.