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

Florian Luca ¹   ; Augustine O. Munagi ²

¹ Mathematical Institute UNAM Juriquilla Juriquilla, 76230 Santiago de Querétaro Querétaro de Arteaga, México and School of Mathematics University of the Witwatersrand P.O. Box Wits 2050 Johannesburg, South Africa
² The John Knopfmacher Centre for Applicable Analysis and Number Theory University of the Witwatersrand P.O. Box Wits 2050 Johannesburg, South Africa

Colloquium Mathematicum, Tome 134 (2014) no. 2, pp. 193-209

Résumé

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 $m,m_1,m_2$.

Détail

DOI : 10.4064/cm134-2-4

Keywords: note every positive integer has representation sum distinct members sequence where number divisors member binary recurrence satisfying mild technical conditions number summands tends infinity rate least log log log positive constant compute fibonacci numbers form positive integers