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On The Number of Binomial Coefficients Which are Divisible by Their Row Number: II
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- Journal:
- Canadian Mathematical Bulletin / Volume 28 / Issue 4 / 01 December 1985
- Published online by Cambridge University Press:
- 20 November 2018, pp. 481-486
- Print publication:
- 01 December 1985
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If n is a natural number, let A(n) denote the number of integers, k, such that 0 < k < n and n divides
. Let ϕ(n) denote Euler's totient function. Necessary and sufficient conditions are given so that A(n) = ϕ(n) when n is square-free.
. Let ϕ(On the Number of Binomial Coefficients which are Divisible by their Row Number
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- Journal:
- Canadian Mathematical Bulletin / Volume 25 / Issue 3 / 01 September 1982
- Published online by Cambridge University Press:
- 20 November 2018, pp. 363-365
- Print publication:
- 01 September 1982
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If n is a natural number, let A(n) be the number of integers, k, such that 0 < k < n and n divides
. Then ϕ(n) ≤ A(n) ≤ n - 1 - 2ω(n) + ɛ, where ?(n) denotes the number of distinct prime factors of n, and ɛ = 0 unless n is twice a prime, in which case ɛ = 1.
. Then ϕ(Some Generalizations of an Identity of Subhankulov
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- Journal:
- Canadian Mathematical Bulletin / Volume 20 / Issue 4 / 01 December 1977
- Published online by Cambridge University Press:
- 20 November 2018, pp. 489-494
- Print publication:
- 01 December 1977
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In 1957, M. A. Subhankulov established the following identity
where
; μ is the Môbius function and J2 is the Jordan totient function of order 2. Since the Ramanujan trigonometrical sum C(nr) = ∑d| (n, r)dμ(r/d), we rewrite the above identity using C(n, r).In this paper, we give a generalization of Ramanujan's sum, which generalizes some of the earlier generalizations mainly due to E. Cohen, and prove a theorem from which we deduce some generalizations of the above identity.
; μ is the Môbius function and 