The Semiprimorial Triad: First Prime Offsets, First Semiprime Offsets, and Semiprimes at +1

A semiprimorial is the product of the first n semiprimes. This paper states three conjectures about what happens when positive integers are added to a semiprimorial: the first offset that reaches a prime is always prime, the first offset that reaches a semiprime is always prime, and adding 1 produces a semiprime infinitely often. An elementary divisibility theorem explains why the first two questions are related. Computation verifies the two first-offset conjectures through n = 100 and rigorously certifies thirteen semiprime values of Q_n + 1 in the same range, with six cases still unresolved.