ABSTRACT
In some rather embarrassing cases, such as the odd perfect numbers, odd weirds, and crowds, we do not even known whether the sets possess any members at all, although they could quite easily contain an infinite number. Such is the degree of our ignorance. This chapter focuses on these kinds of difficulties by setting up even more simply defined sequences of integers and asking once again whether these series go on forever, or whether at some point they stop. In some, as is the case with the perfect and amicable numbers, the answers remain unknown, while in others some insight has been achieved. The idea is to start with any integer whatsoever and to generate a series in which each term after the first is the sum of the divisors of the preceding term. The chapter concludes with brief reference to a slightly different, but related, class of unsolved sequence problems called 'looping problems'.
