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On integers with a special divisibility property. (English) Zbl 1164.11050

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 \(p<q\), and \(a\), \(b\) are positive integers. If \(n\neq 2997\), then \(n\in {\mathcal B}\) if and only if \(b=1\) and there exists a positive integer \(k\) such that \(q=2p^{(p^k-1)/(p-1)}+1\) and \(a=k+2(p^k-1)/(p-1)\).
The authors remark that they do not have any conjecture about the correct order of magnitude of \(\#{\mathcal B}(x)\) as \(x\to \infty \) and that they cannot even show that \({\mathcal B}\) is an infinite set, though Hardy-Littlewood conjectures suggest that the inequality \(\#{\mathcal B}(x)\gg \sqrt x/(\log x)^2\) should hold true.

MSC:

11N37 Asymptotic results on arithmetic functions
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