Some Groupoids and their Representations by Means of Integer Sequences

Author(s): Amelia Carolina Sparavigna

In some previous works, we have discussed the groupoids related to the integer sequences of Mersenne, Fermat, Cullen, Woodall and other numbers. These groupoids possess different binary operators. As we can easily see, other integer sequences can have the same binary operators, and therefore can be used to represent the related groupoids. Using the On-Line Encyclopedia of Integer Sequences (OEIS), we are able to identify the properties of these representations of groupoids. At the same time, we can also find integer sequences not given in OEIS and probably not yet studied.

Groupoid Representations, Integer Sequences, Binary Operators, Generalized Sums, Generalized Entropies, Tsallis Entropy, Q-Calculus, Abelian Groups, Fermat Numbers, Mersenne Numbers, Triangular Numbers, Repunits, Oblong Numbers

1. Stover, Christopher and Weisstein, Eric W. "Groupoid." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Groupoid.html
2. Sparavigna, A. C. (2019). Composition Operations of Generalized Entropies Applied to the Study of Numbers, International Journal of Sciences, Vol. 8, Issue 4, pp. 87-92. DOI: 10.18483/ijSci.2044
3. Sparavigna, A. C. (2019). Binary Operators of the Groupoids of OEIS A093112 and A093069 Numbers (Carol and Kynea Numbers). Zenodo. June 6. DOI: 10.5281/zenodo.3240465
4. Sparavigna, A. C. (2019). Groupoids of OEIS A002378 and A016754 Numbers (oblong and odd square numbers). Zenodo. June 16. DOI: 10.5281/zenodo.3247003
5. Sparavigna, A. C. (2019. Groupoid of OEIS A001844 Numbers (centered square numbers). Zenodo. June 22. DOI: 10.5281/zenodo.3252339
6. Sparavigna, A. C. (2019). Groupoid of OEIS A003154 Numbers (star numbers or centered dodecagonal numbers). Zenodo. Septembr 5. DOI: 10.5281/zenodo.3387054
7. Sparavigna, A. C. (2019,. The groupoid of the Triangular Numbers and the generation of related integer sequences. Zenodo, October 2. DOI: 10.5281/zenodo.3470205
8. Sparavigna, A. C. (2018, May 20). On a generalized sum of the Mersenne Numbers. Zenodo, May 20. DOI: 10.5281/zenodo.1250048
9. Sparavigna, A. C. (2018). The q-Integers and the Mersenne Numbers. SSRN Electronic Journal, May 23. DOI: 10.2139/ssrn.3183800
10. Tsallis, C. (1988). Possible Generalization of Boltzmann-Gibbs Statistics, Journal of Statistical Physics, Vol. 52, pp. 479–487. DOI:10.1007/BF01016429
11. Curado, E. M., Tempesta, P., & Tsallis, C. (2016). A new entropy based on a group-theoretical structure. Annals of Physics, Vol. 366, pp. 22-31. DOI: 10.1016/j.aop.2015.12.008
12. Sparavigna, A. C. (2019). On Repunits. Zenodo, April 14. DOI: 10.5281/zenodo.2639620
13. Sparavigna, A. C. (2018). The group of the Fermat Numbers. Zenodo, May 24. DOI: 10.5281/zenodo.1252422
14. Weisstein, Eric W. "Fermat Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/FermatNumber.html
15. Sparavigna, A. C. (2019). On the generalized sums of Mersenne, Fermat, Cullen and Woodall Numbers. Zenodo, April 9. DOI: 10.5281/zenodo.2634312
16. https://oeis.org/A093112
17. https://oeis.org/A093069