Some problems of Erdős on the sum-of-divisors function

Abstract

Abstract. Let σ(n) denote the sum of all of the positive divisors of n, and let s(n) = σ(n) − n denote the sum of the proper divisors of n. The functions σ(·) and s(·) were favorite subjects of investigation by the late Paul Erdős. Here we revisit three themes from Erdős's work on these functions. First, we improve the upper and lower bounds for the counting function of numbers n with n deficient but s(n) abundant, or vice versa. Second, we describe a heuristic argument suggesting the precise asymptotic density of n not in the range of the function s(·); these are the so-called nonaliquot numbers. Finally, we prove new results on the distribution of friendly k-sets, where a friendly k-set is a collection of k distinct integers which share the same value of σ(n) n .

Perfect Numbers With Identical Digits in Negative Base

“…There are many open problems concerning the function σ (e.g., see [11]), and we do not even know whether there exist infinitely many perfect numbers. Recently, R. B. Nelsen [9] proved that every even perfect number ends in 6 In this short note we are interested in those positive integers which are repdigits in base b and perfect.…”

Finite connected components of the aliquot graph

“…Recall that a number n ∈ N is called non-aliquot (or untouchable) if s −1 ({n}) = ∅. Pollack and Pomerance [11] have conjectured that the non-aliquot numbers have asymptotic density in the natural numbers, and this is supported by the available numerical evidence. Their analysis relies heavily on some heuristics for the typical behavior of s over the natural numbers.…”

Numerical and Statistical Analysis of Aliquot Sequences

“…First, in Section 2 we describe our computation of the geometric mean µ k (X = 2 37 ) of s k (n)/s k−1 (n) taken over all even n ≤ 2 37 with s k (n) > 0 for 1 ≤ k ≤ 10. In Section 4, we describe the enumeration of untouchable numbers (those not in the range of s(n)), extending the table of Pollack and Pomerance [2016] from a bound of 10 10 to 2 40 . We use these data to compute µ 1 (2 40 ) taken over even n that are in the range of s(n).…”