Class numbers, Ono invariants and some interesting primes

Our aim is to find all the prime numbers such that has at most two different prime factors, for all the odd integers such that . We solve entirely the cases , using the knowledge of the quadratic imaginary number fields with class numbers 4, 1 and 2 respectively. The case is not completely solved. Taking into account a result of Stéphane Louboutin, we prove that there is at most one value besides our list. Assuming a Restricted Riemann Hypothesis, the list is complete. In the last section of the paper we give a short sketch for the general problem: find all odd integers such that has at most two different prime factors, for all the odd integers such that .