^{1}

^{1}

^{2}

^{1}

^{2}

Let a

Prime numbers, the building blocks of any positive integer, fascinate math lovers [

Since 300 BC, the irregular distribution of primes throughout the sequence of natural numbers has been extensively investigated. Giants as Chebyshev, Dirichlet, Eratosthenes, Erdös, Euclid, Euler, Fermat, Gauss, Legendre, and Riemann analyzed this matter [

Assume that a

The aim of this study based on

Let a

Let ^{th}

Sequences of

2 | |

3 | |

4 | |

5 | |

6 | |

7 | |

8 | |

9 | |

10 | |

11 |

Sequences of first-order differences of consecutive

2 | |

3 | |

4 | |

5 | |

6 | |

7 | |

8 | |

9 | |

10 | |

11 |

In order to evaluate the variability of the series

The entropy

As the next step, the time series

Consider that

Thus, these nodes are connected if there is a straight line joining

In the horizontal visibility (HV) graph [

Numerical experiments were performed by taking

The normalized entropy

Normalized informational entropy

2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | |
---|---|---|---|---|---|---|---|---|---|---|

0.713 | 0.997 | 0.653 | 1.000 | 0.797 | 1.000 | 0.696 | 0.999 | 0.700 | 1.000 |

From the gap sequences

Figures

The degree distribution found in the NV graphs (dots) and the corresponding fitted function

The degree distribution found in the HV graphs (dots) and the corresponding fitted function

Tables

Values of

Visibility | ||||
---|---|---|---|---|

Natural | 2 | 6.17 | ||

Natural | 3 | 6.52 | ||

Natural | 4 | 6.40 | ||

Natural | 5 | 14.3 | ||

Natural | 6 | 6.33 | ||

Natural | 7 | 27.7 | ||

Natural | 8 | 6.28 | ||

Natural | 9 | 6.66 | ||

Natural | 10 | 6.23 | ||

Natural | 11 | 45.2 |

Values of

Visibility | ||||
---|---|---|---|---|

Horizontal | 2 | 3.67 | ||

Horizontal | 3 | 3.98 | ||

Horizontal | 4 | 3.47 | ||

Horizontal | 5 | 3.97 | ||

Horizontal | 6 | 3.89 | ||

Horizontal | 7 | 3.96 | ||

Horizontal | 8 | 3.59 | ||

Horizontal | 9 | 3.99 | ||

Horizontal | 10 | 3.84 | ||

Horizontal | 11 | 3.95 |

By reducing the quantity of

From the numerical experiments reported in this work, the following relations were found:

The absence of a discernible pattern in the sequence of prime numbers has historically hampered the derivation of a formula for correctly generating such a sequence [

As shown in Table

As shown in Figures

It is well known that, for HV graphs obtained for periodic sequences of period

A final comment about the generation of

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

BLM thanks to Instituto Presbiteriano Mackenzie for the scholarship. LHAM is partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) under grant #304081/2018-3. This study was financed in part by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)—finance code 001.