Find $p$ such that $(p-1)$ has many small divisors

Question

I'm looking for a prime $p$ such that $(p-1)$ has many "small", preferably distinct, divisors.

I tried framing the question as solutions for $p$ to the system, \begin{align*} \phi(p) = 0 \mod p_i \quad \text{for } i=1\cdots n \end{align*} where $p_i$ is the i'th prime.

Does anybody know any efficient ways to solve this or have some literature that touches this topic?

Answer 1

You can't do much better than http://oeis.org/A018239, which has the primes that are one more than a primorial. In particular $200560490131$ is one more than the product of all the primes up to $31$. The next one is rather large $$171962010545840643348334056831754301958457563\\ 589574256043877110505832165523856261308397965147\\ 9555788009994557822024565226932906295208262756822\\ 275663694111$$ You can find more from http://oeis.org/A005234 which gives the highest prime to multiply before adding $1$ to get a prime.