On Positive Numbers U For
نویسنده
چکیده
Let n be a positive integer n and let uj(n), 0 (n) 3 r (n) ? (n) and a(n) be the classical arithmetic functions of n. That is, cj(n), fi(n), and r(n) count the number of distinct prime divisors of n, the total number of prime divisors of n, and the number of divisors of n, respectively, while 4>(n) and a(n) are the Euler function of n and the sum of divisors function of n respectively. A lot of interest has been expressed in investigating the asymptotic densities of the sets of n for which one of the "small' arithmetic functions of n divides some other arithmetic function of n. For example, in [2], it was shown that the set of n for which uj(n) divides n is of asymptotic density zero. This result was generalized in [4]. The formalism from [4] implies, in particular, that the set of n for which either Q(n) or r(n) divide n is also of asymptotic density zero. On the other hand, in [1] it is shown that r(n) divides a(n) for almost all n and, in fact, it can be shown that all three numbers u)(n), ft(n) and r(n) divide both 0(n) and o~(n) for almost all n. In this note, we look at the set of positive integers n for which one of the small arithmetic functions of n divides Fn or Ln. Here, Fn and Ln are the TI*̂ 1 Fibonacci numbers and the JI^^ Lucas number, respectively. We have the following result:
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