Expansions of binary recurrences in the additive base formed by the number of divisors of the factorial
Volume 134 / 2014
Colloquium Mathematicum 134 (2014), 193-209
MSC: Primary 11B39; Secondary 11D72, 11N37.
DOI: 10.4064/cm134-2-4
Abstract
We note that every positive integer $N$ has a representation as a sum of distinct members of the sequence $\{d(n!)\}_{n\ge 1}$, where $d(m)$ is the number of divisors of $m$. When $N$ is a member of a binary recurrence ${\bf u}=\{u_n\}_{n\ge 1}$ 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 $m,m_1,m_2$.
