Caldwell's conjecture for the distribution of Cunningham chain lengths

Fix a natural number $k$. For each prime $p$, let $l(p)$ denote the length of its Cunningham chain, and for each prime $q$ let $\operatorname{ord}(q;2)$ be the multiplicative order of $2$ modulo $q$. Define

$$B_k=2^{k-1}\prod_{q>2}\frac{1-\min\{k,\operatorname{ord}(q;2)\}/q}{(1-q^{-1})^k}.$$

Caldwell's conjecture. The number of primes $p\leq N$ with $l(p)\geq k$ satisfies

$$\sum_{\substack{p\leq N\\ l(p)\geq k}}1\sim B_k\int_2^N\frac{dx}{(\log x)(\log 2x)\cdots(\log 2^{k-1}x)}\sim B_k\frac{N}{\log^kN}.$$

This is the Bateman–Horn prediction specialized to Cunningham chains; the paper presents it as an expectation and uses it to study the maximal chain length.