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On Representations as a Sum of Consecutive Integers

  • W. J. Leveque (a1)
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1. Introduction. It is the object of this paper to investigate the function γ(m), the number of representations of m in the form

(1)

where . It is shown that γ(m) is always equal to the number of odd divisors of m, so that for example γ(2 k ) = 1, this representation being the number 2 k itself. From this relationship the average order of γ(m) is deduced ; this result is given in Theorem 2. By a method due to Kac [2], it is shown in §3 that the number of positive integers for which γ(m) does not exceed a rather complicated function of n and ω, a real parameter, is asymptotically nD(ω), where D(ω) is the probability integral

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References
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[1] Hardy, G. H. and Wright, E. M., An introduction to the theory of numbers (Oxford, 1945).
[2] Kac, M., Note on the distribution of values of the arithmetic function d(m), Bulletin Amer. Math. Soc, vol. 47 (1941), 815817.
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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
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