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ON A SEQUENCE INVOLVING SUMS OF PRIMES
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 88 / Issue 2 / October 2013
- Published online by Cambridge University Press:
- 18 January 2013, pp. 197-205
- Print publication:
- October 2013
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- Article
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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.
