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- Title
SOME PROBLEMS OF ERDŐS ON THE SUM-OF-DIVISORS FUNCTION.
- Authors
POLLACK, PAUL; POMERANCE, CARL
- 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 Erd˝os. Here we revisit three themes from Erd˝os'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.
- Subjects
ERDOS, Paul, 1913-1996; INTEGERS; HEURISTIC algorithms
- Publication
Transactions of the American Mathematical Society, Series B, 2016, Vol 3, Issue 1, p1
- ISSN
2330-0000
- Publication type
Article
- DOI
10.1090/btran/10