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ON A SEQUENCE INVOLVING SUMS OF PRIMES

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

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. 

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