On the Difference Between an Integer and the Sum of its Proper Divisors
Department or Administrative Unit
Let σ(n) be the sum of the divisors of n. Although much attention has been paid to the possible values of σ(n)−n (the sum of proper divisors), comparatively little work has been done on the possible values of e(n):=σ(n)−2n. Here we present some theoretical and computational results on these values. In particular, we exhibit some infinite and possibly infinite families of integers that appear in the image of e(n). We also find computationally all values of n<1020 for which e(n) is odd, and we present some data from our computations. At the end of this paper, we present some conjectures suggested by our computational work.
Davis, N., Klyve, D. & Kraght, N. (2013). On the difference between an integer and the sum of its proper divisors. Involve, 6(4), 493-504. DOI: 10.2140/involve.2013.6.493
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