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ON A SEQUENCE INVOLVING SUMS OF PRIMES

Part of: Elementary number theory Diophantine approximation, transcendental number theory Sequences and sets Additive number theory; partitions Combinatorics

Published online by Cambridge University Press:  18 January 2013

ZHI-WEI SUN
ZHI-WEI SUN*
Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, PR China email [email protected]
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Abstract

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For $n= 1, 2, 3, \ldots $ let ${S}_{n} $ be the sum of the first $n$ primes. We mainly show that the sequence ${a}_{n} = \sqrt[n]{{S}_{n} / n}~(n= 1, 2, 3, \ldots )$ is strictly decreasing, and moreover the sequence ${a}_{n+ 1} / {a}_{n} ~(n= 10, 11, \ldots )$ is strictly increasing. We also formulate similar conjectures involving twin primes or partitions of integers.

Keywords

primessums of primesmonotonicitytwin primespartitions of integers

MSC classification

Secondary: 11A41: Primes 05A17: Partitions of integers 05A20: Combinatorial inequalities 11B75: Other combinatorial number theory 11B83: Special sequences and polynomials 11J99: None of the above, but in this section 11P83: Partitions; congruences and congruential restrictions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 88 , Issue 2 , October 2013 , pp. 197 - 205
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

References

Abramowitz, M. and Stegun, I. A. (eds.) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing (Dover, New York, 1972).Google Scholar
Chen, W. Y. C., Recent developments on log-concavity and $q$-log-concavity of combinatorial polynomials. Paper presented at the 22nd Int. Conf. on Formal Power Series and Algebraic Combin. (San Francisco, 2010).Google Scholar
Dusart, P., ‘The $k\mathrm{th} $ prime is greater than $k(\log k+ \log \log k- 1)$ for $k\geqslant 2$’, Math. Comp. 68 (1999), 411–415.CrossRefGoogle Scholar
Hardy, G. H. and Ramanujan, S., ‘Asymptotic formulae in combinatorial analysis’, Proc. Lond. Math. Soc. 17 (1918), 75–115.CrossRefGoogle Scholar
Janoski, J. E., A Collection of Problems in Combinatorics, PhD Thesis, Clemson University, May 2012.Google Scholar
Liu, L. L. and Wang, Y., ‘On the log-convexity of combinatorial sequences’, Adv. in Appl. Math. 39 (2007), 453–476.CrossRefGoogle Scholar
Ribenboim, P., The Little Book of Bigger Primes, 2nd edn (Springer, New York, 2004).Google Scholar
Rosser, J. B., ‘The $n$th prime is greater than $n\log n$’, Proc. Lond. Math. Soc. 45 (1939), 21–44.CrossRefGoogle Scholar
Rosser, J. B. and Schoenfeld, L., ‘Approximate formulas for some functions of prime numbers’, Illinois J. Math. 6 (1962), 64–94.CrossRefGoogle Scholar
Sun, Z.-W., ‘On functions taking only prime values’, Preprint, 2012, arXiv:1202.6589.Google Scholar