Finite connected components of the aliquot graph

Abstract

Conditional on a strong form of the Goldbach conjecture, we determine all finite connected components of the aliquot graph containing a number less than 10 9 , as well as those containing an amicable pair below 10 14 or one of the known perfect or sociable cycles below 10 17 . Along the way we develop a fast algorithm for computing the inverse image of an even number under the sum-of-proper-divisors function.

Divisor‐sum Fibers

“…Our proof of Theorem 1.2 is presented in §2. We borrow some ideas from recent work of Booker [3]; Booker's arguments by themselves almost immediately give the slightly weaker result with x 1/2+ǫ(x) replaced by x 1/2−ǫ for any fixed (constant) ǫ > 0. EGPS [7, p. 170] point out that their Conjecture 1.1 would be a consequence of the following assertion about the sizes of elements in a fiber.…”