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3 results

  • ON A SEQUENCE INVOLVING SUMS OF PRIMES

  • Part of
  • ZHI-WEI SUN
  • 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
    • 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.

  • Oil exploration: sequential decisions in the face of uncertainty

  • L. Benkherouf, J. A. Bather
  • Journal:
    Journal of Applied Probability / Volume 25 / Issue 3 / September 1988
    Published online by Cambridge University Press:
    14 July 2016, pp. 529-543
    Print publication:
    September 1988

  • A simple Bayesian model for oil exploration is suggested to investigate strategies for drilling. A condition on the way successes and failures affect the prior distribution implies a certain form of the detection mechanism. It is shown that the problem of finding strategies for drilling reduces to an optimal stopping problem. Two new families of distributions are obtained with generating functions related to classical work on partitions of integers. By using such distributions and simple mixtures of them as priors, the stopping problem can be solved explicitly. This leads to the construction of simple strategies and their effectiveness is demonstrated by evaluating suitable operating characteristics.