On the divisors of natural and happy numbers: a study based on entropy and graphs

B.L. Mayer; L.H.A. Monteiro; B.L. Mayer; L.H.A. Monteiro

doi:10.3934/math.2023679

AIMS Mathematics

2023, Volume 8, Issue 6: 13411-13424. doi: 10.3934/math.2023679

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On the divisors of natural and happy numbers: a study based on entropy and graphs

B.L. Mayer ¹,
L.H.A. Monteiro ^{1,2
,
,}

1.
Universidade Presbiteriana Mackenzie, Escola de Engenharia, São Paulo, SP, Brazil
2.
Universidade de São Paulo, Escola Politécnica, São Paulo, SP, Brazil

Received: 03 January 2023 Revised: 19 March 2023 Accepted: 27 March 2023 Published: 06 April 2023
MSC : 05C07, 11A51, 40A05

The features of numerical sequences and time series have been studied by using entropies and graphs. In this article, two sequences derived from the divisors of natural numbers are investigated. These sequences are obtained either directly from the divisor function or by recursively applying the divisor function. For comparison purposes, analogous sequences formed from the divisors of happy numbers are also examined. Firstly, the informational entropy of these four sequences is numerically determined. Then, each sequence is mapped into graphs by employing two visibility algorithms. For each graph, the average degree, the average shortest-path length, the average clustering coefficient, and the degree distribution are calculated. Also, the links in these graphs are quantified in terms of the parity of the numbers that these links connect. These computer experiments suggest that the four analyzed sequences exhibit characteristics of quasi-random sequences.
- divisor function,
- happy number,
- informational entropy,
- natural number,
- visibility graph
Citation: B.L. Mayer, L.H.A. Monteiro. On the divisors of natural and happy numbers: a study based on entropy and graphs[J]. AIMS Mathematics, 2023, 8(6): 13411-13424. doi: 10.3934/math.2023679

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Abstract

The features of numerical sequences and time series have been studied by using entropies and graphs. In this article, two sequences derived from the divisors of natural numbers are investigated. These sequences are obtained either directly from the divisor function or by recursively applying the divisor function. For comparison purposes, analogous sequences formed from the divisors of happy numbers are also examined. Firstly, the informational entropy of these four sequences is numerically determined. Then, each sequence is mapped into graphs by employing two visibility algorithms. For each graph, the average degree, the average shortest-path length, the average clustering coefficient, and the degree distribution are calculated. Also, the links in these graphs are quantified in terms of the parity of the numbers that these links connect. These computer experiments suggest that the four analyzed sequences exhibit characteristics of quasi-random sequences.

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