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

• Autorzy: Florian Luca , Augustine O. Munagi
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 11N37: Asymptotic results on arithmetic functions
• 11B39: Fibonacci and Lucas numbers and polynomials and generalizations
• 11D72: Equations in many variables

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2014
• Tom: 134
• Numer: 2
• Strony: 193-209

Daty

wydano

2014

Twórcy

autor

Florian Luca

• Mathematical Institute, UNAM Juriquilla, Juriquilla, 76230 Santiago de Querétaro, Querétaro de Arteaga, México
• School of Mathematics, University of the Witwatersrand, P.O. Box Wits 2050, Johannesburg, South Africa

autor

Augustine O. Munagi

• The John Knopfmacher Centre, for Applicable Analysis and Number Theory, University of the Witwatersrand, P.O. Box Wits 2050, Johannesburg, South Africa

Identyfikatory

DOI

10.4064/cm134-2-4