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Iterating the sum-of-divisors function. (English) Zbl 0866.11003
Let \(\sigma\) be the sum-of-divisors function and define \(\sigma^m(n)= \sigma(\sigma^{m-1}(n))\) for \(m>1\). A number \(n\) is said to be \((m,k)\)-perfect if \(\sigma^m(n)=kn\), so that the classical perfect numbers are \((1,2)\)-perfect. The authors give a table of all \((2,k)\)-perfect numbers up to \(10^9\); for example \(2^5\cdot 3^2\cdot 5\cdot 7^2\cdot 11\cdot 13\cdot 31\) is \((2,19)\)-perfect.
Besides the notorious problem of whether there are infinitely many perfect numbers, one may now ask whether every number is \((m,k)\)-perfect for some suitable values of \(m\) and \(k\). The authors have verified that this is so for \(n\leq 1000\), and some results for \(n\leq 400\) are given. For example, \(n=389\) is \((m,k)\)-perfect with the least value \(m=296\) and \(k= 2^{93}\cdot 3^{10}\cdots\approx 5\cdot 10^{232}\). There are 14 values for \(n\leq 400\) whose corresponding least values for \(m\) exceed those for \(n\), namely \(n=3,11,29,53, 58,59,67, 101,109,131, 149,173, 202,239\). Naturally, the computations for such cases are relatively more complicated.
It had been suggested that, corresponding to any \(n_1,n_2\), there are \(m_1,m_2\) such that \(\sigma^{m_1}(n_1)= \sigma^{m_2}(n_2)\). However, from their computational evidence, the authors believe that this may be false.
11A25 Arithmetic functions; related numbers; inversion formulas
11Y70 Values of arithmetic functions; tables
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