Journal of Integer Sequences, Vol. 24 (2021), Article 21.8.5

On a Sequence Related to the Factoradic Representation of an Integer

Maximiliano Sánchez Garza
Facultad de Ciencias Físico Matemáticas
Universidad Autónoma de Nuevo León
Av. Universidad
Ciudad Universitaria
San Nicolás de los Garza, Nuevo León 66451

Enrique Treviño
Mathematics Department
Lake Forest College
555 N. Sheridan Road
Lake Forest, IL 60045


For a positive integer r, define jr to be the smallest positive integer n satisfying n! > nr–1. In this paper we prove jr+1 ∈ {jr + 1, jr + 2}, which leads us to explore the set of positive integers r for which jr+1 = jr + 2. We prove this set has the same density as the prime numbers. The sequence jr was introduced by Carlson, Goedhart, and Harris in their work on factoradic happy numbers, and we prove some properties of jr that lead to an improvement of one of their theorems.

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(Concerned with sequences A007770 A027641 A230319 A336803.)

Received January 26 2021; revised versions received January 27 2021; July 20 2021. Published in Journal of Integer Sequences, August 17 2021.

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