Characterize when the first nonintegral index is composite

Determine why the value N_k, defined as the smallest positive integer n for which the nth term of the k-Göbel sequence is not an integer, is prime for most values of k, and characterize the cases in which N_k is composite.

Background

For each integer k ≥ 2, the k-Göbel sequence is defined by g_{k,1}=2 and n g_{k,n}=(n−1)g_{k,n−1}+g_{k,n−1}k. The quantity N_k is the first index at which this sequence ceases to be integral. The paper explains that the behavior of the sequence (N_k)_k is poorly understood and identifies the primality of N_k as one of three fundamental problems. Numerical evidence reported in the paper shows that N_k is prime for 86.5% of values with k ≤ 107, but no general characterization is known.

References

The behavior of the sequence (N_k)_k ([7, A108394]) is not yet understood well and remains mysterious. In [6], Matsuhira, Matsusaka, and Tsuchida proved that mink≥2. Nk = 19. As mentioned in [6, Section 3] and [4, Episode 3], the following three questions are fundamental problems about the sequence (Nk)k: (1) Why is Nk a prime number for most values of k? Or rather, in what cases does Nk become a composite number?

— On the length over which $k$-Göbel sequences remain integers  (2502.17448 - Kobayashi et al., 5 Feb 2025) in Section 1, Introduction, page 1

In [6, Section 3] and [4, Episode 3], the following three questions are fundamental problems about the sequence (Nk)k: (1) Why is Nk a prime number for most values of k? Or rather, in what cases does Nk become a composite number?

— On the length over which $k$-Göbel sequences remain integers  (2502.17448 - Kobayashi et al., 5 Feb 2025) in Section 1, Introduction, p. 1