General limit formulae involving prime numbers
Abstract
Let pn be the nth prime number. In this note, we study strictly increasing sequences of positive integers An such that the limit limn→∞(A1A2···An)1/pn = e holds. This limit formula is in fact a generalization of some previously known results. Furthermore, some other generalizations are established.
Keywords
limit formula; generalization; the number e; prime numbers
References
R. Farhadian, A remark on lim_{n→∞}sqrt[p_n]{p_1p_2···p_n} = e, Math. Gaz. 105 (2021), 311–312.
R. Farhadian, A generalization of Euler’s limit, Amer. Math. Monthly 129 (2022), 384.
R. Farhadian and R. Jakimczuk, Notes on a general sequence, Ann. Math. Sil. 34 (2020), 193–202.
R. Farhadian and R. Jakimczuk, A note on the geometric mean of prime numbers and generalizations, J. Discrete Math. Sci. Cryptogr. (2020). DOI: 10.1080/09720529.2020.1723920
R. Farhadian and R. Jakimczuk, A note on two fundamental recursive sequences, Ann. Math. Sil. 35 (2021), 172–183.
R. Farhadian and R. Jakimczuk, A further generalization of lim_{n→∞}sqrt[n]{n!}/n = 1/e, Ann. Math. Sil. (2022). DOI: 10.2478/amsil-2022-0006
R. Farhadian and R. Jakimczuk, On a sequence involving products of primes, J. Interdiscip. Math. (2022). DOI: 10.1080/09720502.2021.1961981
S.R. Finch, Mathematical Constants, Cambridge University Press, Cambridge, 2003.
R. Jakimczuk, The ratio between the average factor in a product and the last factor, Math. Sci. Q. J. 1 (2007), 53–62.
R. Jakimczuk, Functions of slow increase and integer sequences, J. Integer Seq. 13 (2010), Article 10.1.1, 14 pp.
J. Rey Pastor, P. Pi Calleja, and C.A. Trejo, Análisis Matemático, Vol. 1, Editorial Kapelusz, Buenos Aires, 1969.
Department of Statistics, Razi University, Iran Iran, Islamic Republic of
https://orcid.org/0000-0003-4027-9838
División Matemática, Universidad Nacional de Luján, Argentina Argentina
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