Sequences in finite fields yielding divisors of Mersenne, Fermat and Lehmer numbers, II

Let ρ be an odd prime ≥ 11. In Part I, starting from an M-cycle in a finite field F_ρ, we have established how the divisors of Mersenne, Fermat and Lehmer numbers arise. The converse question is taken up in this Part with the introduction of an arithmetic function and the notion of a split-associated prime.