Minimal multiplier for realizability of Stirling-number sequences

Ascertain, for each integer k ≥ 1, the minimal constant multiplier C_k such that the sequence S_k = (Stirling numbers of the second kind S(n+k−1, k)) becomes realizable; specifically, determine whether finite computations of the least common multiple of the denominators in the sequence (1/n)∑_{d|n} μ(n/d) S(d+k−1, k) reach (k−1)! or otherwise provide the exact minimal value C_k.

Background

For fixed k, define S_k as the sequence built from Stirling numbers of the second kind. It is known that S_k is realizable for k=2, and for k≥3 it is not realizable but is almost realizable with a minimal constant multiplier C_k that divides (k−1)!.

The minimal multiplier C_k is defined via the least common multiple of denominators in an infinite sequence of rational numbers derived from Möbius-weighted sums. Verifying minimality through finite computation is problematic, and the exact values of C_k are not currently determined.