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

Florian Luca, and Augustine O. Munagi. "Expansions of binary recurrences in the additive base formed by the number of divisors of the factorial." Colloquium Mathematicae 134.2 (2014): 193-209. <http://eudml.org/doc/283859>.

@article{FlorianLuca2014,
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 $u = \{uₙ\}_\{n≥1\}$ satisfying some mild technical conditions, we show that the number of such summands tends to infinity with n at a rate of at least 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.},
author = {Florian Luca, Augustine O. Munagi},
journal = {Colloquium Mathematicae},
keywords = {binary recurrences; factorials; applications of linear forms in logarithms},
language = {eng},
number = {2},
pages = {193-209},
title = {Expansions of binary recurrences in the additive base formed by the number of divisors of the factorial},
url = {http://eudml.org/doc/283859},
volume = {134},
year = {2014},
}

TY - JOUR
AU - Florian Luca
AU - Augustine O. Munagi
TI - Expansions of binary recurrences in the additive base formed by the number of divisors of the factorial
JO - Colloquium Mathematicae
PY - 2014
VL - 134
IS - 2
SP - 193
EP - 209
AB - 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 $u = {uₙ}_{n≥1}$ satisfying some mild technical conditions, we show that the number of such summands tends to infinity with n at a rate of at least 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.
LA - eng
KW - binary recurrences; factorials; applications of linear forms in logarithms
UR - http://eudml.org/doc/283859
ER -