A Generalization of the Practical Numbers
A positive integer n is practical if every m <= n a can be written as a sum of distinct divisors of n. One can generalize the concept of practical numbers by applying an arithmetic function f to each of the divisors of n and asking whether all integers in a certain interval can be expressed as sums of f(d)'s, where the d's are distinct divisors of n. We will refer to such n as "f-practical". In this paper, we introduce the f-practical numbers for the first time. We give criteria for when all f-practical numbers can be constructed via a simple necessary-and-sufficient condition, demonstrate that it is possible to construct f-practical sets with any asymptotic density, and prove a series of results related to the distribution of f-practical numbers for many well-known arithmetic functions f.
Schwab, Nicholas, and Lola Thompson. 2018. "A Generalization of the Practical Numbers." International Journal of Number Theory 14(5): 1487-1503.
World Scientific Publishing
International Journal of Number Theory
Practical number, Arithmetic function, Panarithmic number, Sum-of-proper-divisors function