Type: Article

Expansions of binary recurrences in the additive base formed by the number of divisors of the factorial

Luca, F; Munagi, AO;

Abstract:

We note that every positive integer N has a representation as a sum of distinct members of the sequence {d(n!)}(n >= 1), where d(m) is the number of divisors of m. When N is a member of a binary recurrence u = {u(n)}(n >= 1) satisfying some mild technical conditions, we show that the number of such summands tends to infinity with n at a rate of at least c(1) log n/log log n for some positive constant c(1). We also compute all the Fibonacci numbers of the form d(m!) and d(m(1)!) + d(m(2))! for some positive integers m, m(1), m(2).