Iterated stable numbers and iterated amicable pairs

This note presents the restricted divisor functions and describes the amicable pairs and their iterations and the iterated stable numbers derived from these new functions. These arithmetic functions are related to (2)-type Mersenne primes 2^p − 1, (2, 3)-type Mersenne primes 2ⁱ · 3^j − 1 and multiperfect numbers. Using Mathematica 11.2, we have tabulated all iterated stable numbers and iterated amicable pairs up to 2²⁰. We establish several interesting arithmetical properties of the iterated restricted divisor functions. The tables obtained suggested many open problems on iterated stable numbers, and iterated amicable pairs from iteration of the restricted divisor functions, some of which are proven here. For any natural number n, we studied the order of its iteration relative to the restricted divisor functions. Moreover, the generating series of the iterated restricted divisor functions are found.