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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.
##### MSC:
 11A25 Arithmetic functions; related numbers; inversion formulas 11A41 Primes 03H05 Nonstandard models in mathematics
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