Some Properties of the Pseudo-smarandache Function
نویسنده
چکیده
Charles Ashbacher [1] has posed a number of questions relating to the pseudo-Smarandache function Z(n). In this note we show that the ratio of consecutive values Z(n + 1)/Z(n) and Z(n − 1)/Z(n) are unbounded; that Z(2n)/Z(n) is unbounded; that n/Z(n) takes every integer value infinitely often; and that the series ∑ n 1/Z(n) is convergent for any α > 1.
منابع مشابه
Some identities involving the near pseudo Smarandache function
For any positive integer n and fixed integer t ≥ 1, we define function Ut(n) = min{k : 1 t + 2 t + · · · + n t + k = m, n | m, k ∈ N + , t ∈ N + }, where n ∈ N + , m ∈ N + , which is a new pseudo Smarandache function. The main purpose of this paper is using the elementary method to study the properties of Ut(n), and obtain some interesting identities involving function Ut(n). In reference [1], ...
متن کاملOn the mean value of the Near Pseudo Smarandache Function
The main purpose of this paper is using the analytic method to study the asymptotic properties of the Near Pseudo Smarandache Function, and give two interesting asymptotic formulae for it.
متن کاملOn a dual of the Pseudo-Smarandache function
This function generalizes many particular functions. For f( k) = k! one gets the Smarandache function, while for f(k) = k(k: 1) one has the Pseudo-Smarandache function Z (see [1], [4-5]). In the above paper [3] we have defined also dual arithmetic functions as follows: Let 9 : N* -+ N* be a function having the property that for each n 2:: 1 there exists at least a k 2:: 1 such that g(k)ln. Let ...
متن کاملOn the mean value of the Pseudo-Smarandache function
For any positive integer n, the Pseudo-Smarandache function Z(n) is defined as the smallest positive integer k such that n | k(k + 1) 2 . That is, Z(n) = min { k : n| + 1) 2 } . The main purpose of this paper is using the elementary methods to study the mean value properties of p(n) Z(n) , and give a sharper asymptotic formula for it, where p(n) denotes the smallest prime divisor of n.
متن کاملOn the mean value of the Pseudo-Smarandache-Squarefree function
For any positive integer n, the Pseudo Smarandache Squarefree function Zw(n) is defined as Zw(n) = min{m : n|mn, m ∈ N}, and the function Z(n) is defined as Z(n) = min { m : n ≤ m(m + 1) 2 , m ∈ N } . The main purpose of this paper is using the elementary methods to study the mean value properties of the function Zw(Z(n)), and give a sharper mean value formula for it.
متن کامل