On twin EP numbers

Document Type : Research Paper

Authors

1 Department of Computer Science, University of California Santa Barbara, CA 93106, USA

2 Department of Mathematics, Faculty of Science, Selçuk University, Konya 42130, Turkey

Abstract

EP numbers were introduced by Estrada and Pogliani in 2008. These are positive integers $E(n)$ defined as the product of $n$ and the sum of the digits of $n$. Estrada and Pogliani suspected that there may be infinitely many twin EP numbers; i.e. those pairs in this sequence that differ by one. It is relatively easy to show that three consecutive EP numbers do not exist, and that no pair $E(n)$ and $E(m)$ can be twins for infinitely many bases $b$.
The main contribution of our work is the result that indeed there are infinitely many twin EP numbers over any base.
The proof is constructive and makes use of elementary properties of natural numbers. The forms of the twin EP numbers presented are derived from continued fractions. The behavior of the series of the reciprocals of twin EP numbers is also considered.

Keywords

Main Subjects


[1] F. T. Adams-Watters and F. Ruskey, Generating functions for the digital sum and other digit counting sequences, J. Integer Seq., 12 (2009).
[2] J,-P. Allouche and J. Shallit, Sums of digits and the Hurwitz zeta function, Analytic number theory (Tokyo, 1988), Lecture Notes in Math., Springer, Berlin, 1990 19–30.
[3] L. E. Bush, An asymptotic formula for the average sum of the digits of integers, Amer. Math. Monthly, 47 (1940) 154–156.
[4] T. Cai, On 2-Niven numbers and 3-Niven numbers, Fibonacci Quart., 34 no. 2 (1996) 118–120.
[5] C. Cooper and R. E. Kennedy, On consecutive Niven numbers, Fibonacci Quart., 31 no. 2 (1993) 146–151.
[6] E. Estrada and L. Pogliani, A new integer sequence based on the sum of the digits of integers, Kragujevac J. Science, 30 (2008) 45–50.
[7] E. Estrada and P. Pereira-Ramos, Spatial “Artistic” Networks: From Deconstructing Integer-Functions to Visual Arts. Complexity, Vol. 2018, Article ID 9893867.
[8] E. Estrada, Integer-digit functions: an example of math-art integration, Math. Intelligencer, 40 no. 1 (2018) 73–78.
[9] H. G. Grundman, Sequences of consecutive n-Niven numbers, Fibonacci Quart., 32 no. 2 (1994) 174–175.
[10] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford University Press, 5th ed., 1980.
[11] D. R. Kaprekar, Problems involving reversal of digits, Scripta Math., 19 (1953) 81–82.
[12] K. H. Rosen, Discrete Mathematics and its Applications, 7th Edition, McGraw-Hill, New York, 2012, 269–270.
[13] M. Schneider and R. Schneider, Digit sums and generating functions, Ramanujan J., 52 no. 2 (2020) 291–302.
[14] J. O. Shallit, On infinite products associated with sums of digits, J. Number Theory, 21 no. 2 (1985) 128–134.
[15] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences (OEIS), http://oeis.org.
[16] A. J. Van der Poorten, An introduction to continued fractions, Diophantine analysis (Kensington, 1985), London Math. Soc. Lecture Note Ser., 109, Cambridge Univ. Press, Cambridge, 1986 99–138.
[17] C. Vignat and T. Wakhare, Finite generating functions for the sum-of-digits sequence, Ramanujan J., 50 no. 3 (2019) 639–684.
Volume 14, Issue 4 - Serial Number 4
December 2025
Pages 261-270
  • Receive Date: 16 July 2023
  • Revise Date: 02 October 2024
  • Accept Date: 04 October 2024
  • Published Online: 17 November 2024