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Keywords:
divisor function; prime number; iterated sequence; internal set theory
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 $\ell ,m\in \mathbb {N}$ we denote by $\delta ^{s,\ell ,m}( n) $ the sequence $$ \underbrace {d^{s}( d^{s}( \ldots d^{s}( d^{s}( n) +\ell ) +\ell \ldots ) +\ell ) }_{m\text {-times}} =\begin {cases} d^{s}( n) & \text {for} \ m=1,\\ d^{s}( d^{s}( n) +\ell ) &\text {for} \ m=2,\\ d^{s}(d^{s}( d^{s}(n) +\ell ) +\ell ) & \text {for} \ m=3, \\ \vdots & \end {cases} $$ We present classical and nonclassical notes on the sequence $ ( \delta ^{s,\ell ,m}( n)) _{m\geq 1}$, where $\ell $, $n$, $s$ are understood as parameters.
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