A variant of happy sequences

Balasubramanian, Kumar; Parkhi, Nandini

doi:10.2140/involve.2026.19.541

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A variant of happy sequences

Kumar Balasubramanian and Nandini Parkhi

Vol. 19 (2026), No. 4, 541–551

DOI: 10.2140/involve.2026.19.541

Abstract

Let $a_{0}, k \in ℕ$ with $k \geq 2$ . We study a certain sequence of positive integers $(a_{n}) = (f_{a_{0}, k} (n)) = (g (n))$ and demonstrate an interesting property of this sequence. For $c_{0} \in ℕ$ , let $(c_{n}) = (g^{n} (c_{0}))$ , where $g^{n} = g \circ \dots \circ g$ is the composite function. In the particular case when $k = 2$ , we show that there is an interesting duality between the sequences $(a_{n})$ and $(c_{n})$ .

Keywords

integer sequences, cycles, happy numbers

Mathematical Subject Classification

Primary: 11Y55

Secondary: 11A63

Milestones

Received: 9 June 2023

Revised: 15 June 2024

Accepted: 27 February 2025

Published: 13 October 2026

Communicated by Kenneth S. Berenhaut

Authors

Kumar Balasubramanian
Department of Mathematics Indian Institute of Science Education and Research Bhopal Bhopal India

Nandini Parkhi
Department of Mathematics Indian Institute of Science Education and Research Bhopal Bhopal India

Vol. 19, No. 4, 2026