Summary: Let \(p\) range over the set of primes and let \(a\) be a non-zero integer. Here we prove that many properties of the divisors of the natural numbers which can be expressed by inequalities are true for the set \(\{p+a\}\) of shifted primes, too. Among other results we obtain estimates for the quantities \[ \begin{gathered} |\{p:p+a\leq x,\;d_m(p+a)\leq (\log x)^{(1-\alpha)\log m} \text{or} d_m(p+a)\geq (\log x)^{(1+\alpha)\log m}\}|,\\ 0<\alpha< 1, \end{gathered} \] (see Theorem 1) and \[ |\{p:p+a\leq x,\;p+a\;\text{has at least one divisor \(d\) such that} y