Divisor Structure of p-1 in Mersenne Prime Exponents

According to the Wagstaff heuristic, the probability that a Mersenne number $M_p = 2^p-1$ is prime mainly depends on the size of the exponent $p$. We investigate whether the secondary arithmetic structure in $p-1$ is linked to noticeable variation among Mersenne prime exponents, motivated by the cyclotomic decomposition of $2^{p-1}-1$. We introduce the normalized divisor-structure parameter $S(p)=\log \tau(p-1)/\log\log p$, which is a scale-invariant way to measure the divisor structure of $p-1$. which is a scale-invariant way to measure the divisor structure of $p-1$. Using the currently known Mersenne prime exponents (excluding small cases), we compare $S(p)$ with nearby prime controls through percentile analysis, stratified conditional likelihood estimation and stratified permutation testing. Across all methods, Mersenne prime exponents tend to exhibit larger values of $S(p)$ within local exponent windows compared to nearby primes of comparable size. The observed effect is moderate and remains stable across different non-parametric tests. It is also consistent with the classical asymptotic behavior. At present, no analytic explanation for the observed phenomenon is known.