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3 results

  • On the Distribution of the Sequence {nd*(n)}

  • H. L. Abbott, M. V. Subbarao
  • Journal:
    Canadian Mathematical Bulletin / Volume 32 / Issue 1 / 01 March 1989
    Published online by Cambridge University Press:
    20 November 2018, pp. 105-108
    Print publication:
    01 March 1989
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  • Let d*(n) denote the number of unitary divisors of the positive integer n. For x > 1, let B(x) denote the number of integers n for which nd*(n) ≦ x. Balasubramanian and Ramachandra proved that there exists a positive constant β such that . In this note we give an explicit expression for β as an infinite product, namely where the product is over all primes p.

  • Cubic and Higher Order Algorithms for π

  • J. M. Borwein, P. B. Borwein
  • Journal:
    Canadian Mathematical Bulletin / Volume 27 / Issue 4 / 01 December 1984
    Published online by Cambridge University Press:
    20 November 2018, pp. 436-443
    Print publication:
    01 December 1984
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  • We show that the theory of elliptic integral transformations may be employed to construct iterative approximations for π of order p (p any prime). Details are provided for two, three and seven. The cubic case proves amenable to surprisingly complete analysis.

  • Beatty Sequences, Continued Fractions, and Certain Shift Operators

  • Kenneth B. Stolarsky
  • Journal:
    Canadian Mathematical Bulletin / Volume 19 / Issue 4 / 01 December 1976
    Published online by Cambridge University Press:
    20 November 2018, pp. 473-482
    Print publication:
    01 December 1976
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  • Let θ = θ(k) be the positive root of θ2 + (k-2)θ-k = 0. Let f(n) = [(n + l)θ]-[nθ] for positive integers n, where [x] denotes the greatest integer in x. Then the elements of the infinite sequence (f(l), f(2), f(3),…) can be rapidly generated from the finite sequence (f(l), f(2),…,f(k)) by means of certain shift operators. For k = 1 we can generate (the characteristic function of) the sequence [nθ] itself in this manner.