The First Function and Its Iterates
نویسنده
چکیده
Let s(n) denote the sum of the positive divisors of n except for n iteself. Discussed since Pythagoras, s may be the first function of mathematics. Pythagoras also suggested iterating s, so perhaps considering the first dynamical system. The historical legacy has left us with some colorful and attractive problems, mostly still unsolved. Yet the efforts have been productive in the development of elementary, computational, and probabilistic number theory. In the context of the Catalan–Dickson conjecture and the Guy–Selfridge counter conjecture, we discuss the geometric mean of the numbers s(s(2n))/s(2n), thus extending recent work of Bosma and Kane. We also discuss the number of integers m with s(m) = n.
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