Czechoslovak Mathematical Journal, Vol. 69, No. 4, pp. 1177-1196, 2019
On a sequence formed by iterating a divisor operator
Bellaouar Djamel, Boudaoud Abdelmadjid, Özen Özer
Received March 12, 2018. Published online September 16, 2019.
Abstract: 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}} = 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, \dots $
We present classical and nonclassical notes on the sequence $ ( \delta^{s,\ell,m}( n)) _{m\geq1}$, where $\ell,n,s$ are understood as parameters.
Keywords: divisor function; prime number; iterated sequence; internal set theory
