Let \(\lambda \) denote the Carmichael function, i.e.\ for a positive integer \(n\) let \(\lambda (n)\) be the largest order of any element in the multiplicative group \((\mathbb Z/n\mathbb Z)^\times \), and let \(b(n)= \sum _{d| n}\lambda (d)\). The subject of the paper is the set \({\mathcal B}\) of all positive integers \(n\) such that \(b(n)\) is a proper divisor of \(n\). For a positive real number \(x\) let \({\mathcal B}(x)=\{n\in {\mathcal B}\mid n\leq x\}\). The authors prove the following upper bound as \(x\to \infty \): \[ \#{\mathcal B}(x)\leq x\exp \bigl (-2^{-1/2}(1+o(1)) \sqrt {\log x\log \log x}\bigr ), \] where \(\log \) denotes the natural logarithm. Moreover they characterize all odd integers \(n\in {\mathcal B}\) having exactly two prime divisors: Suppose that \(n=p^aq^b\), where \(p\) and \(q\) are odd primes with \(p