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On a sequence formed by iterating a divisor operator. (English) Zbl 07144884
Summary: Let \(\mathbb{N}\) be the set of positive integers and let \(s\in\mathbb{N}\). We denote by \(d^s\) the arithmetic function given by \(d^s(n)=(d(n))^s\), where \(d(n)\) is the number of positive divisors of \(n\). Moreover, for every \(l,m\in\mathbb{N}\) we denote by \(\delta^{s,l,m}(n)\) the sequence \[\underbrace{d^s(d^s(\ldots d^s(d^s(n)+l)+l\ldots)+l)}_{m\text{-times}}=\begin{cases}d^s(n)&\text{for }m=1,\\ d^s(d^s(n)+l)&\text{for }m=2,\\ d^s(d^s(d^s(n)+l)+l)&\text{for }m=3,\\ \vdots&\end{cases}\] We present classical and nonclassical notes on the sequence \((\delta^{s,l,m}(n))_{m\geq 1}\), where \(l\), \(n\), \(s\) are understood as parameters.
11A25 Arithmetic functions; related numbers; inversion formulas
11A41 Primes
03H05 Nonstandard models in mathematics
Full Text: DOI
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