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Cambridge University Press, Bulletin of the Australian Mathematical Society, 2(88), p. 197-205, 2013

DOI: 10.1017/s0004972712000986

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On a sequence involving sums of primes

Journal article published in 2012 by Zhi-Wei Sun
This paper is made freely available by the publisher.
This paper is made freely available by the publisher.

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Abstract

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