We note that every positive integer N has a representation as a sum of distinct members of the sequence , where d(m) is the number of divisors of m. When N is a member of a binary recurrence satisfying some mild technical conditions, we show that the number of such summands tends to infinity with n at a rate of at least c₁logn/loglogn for some positive constant c₁. We also compute all the Fibonacci numbers of the form d(m!) and d(m₁!) + d(m₂)! for some positive integers m,m₁,m₂.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm134-2-4,
author = {Florian Luca and Augustine O. Munagi},
title = {Expansions of binary recurrences in the additive base formed by the number of divisors of the factorial},
journal = {Colloquium Mathematicae},
volume = {135},
year = {2014},
pages = {193-209},
zbl = {06285558},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm134-2-4}
}
Florian Luca; Augustine O. Munagi. Expansions of binary recurrences in the additive base formed by the number of divisors of the factorial. Colloquium Mathematicae, Tome 135 (2014) pp. 193-209. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm134-2-4/