On the Pseudo-Smarandache Function
نویسنده
چکیده
Kashihara[2] defined the Pseudo-Smarandache function Z by m(m+l) } Properties of this function have been studied in [1], [2] etc. 1. By answering a question by C. Ashbacher, Maohua Le proved that S(Z(n»-Z(S(n» changes signs infmitely often. Put d s,z (n) = I S(Z(n»-Z(S(s» I We will prove first that lim inf d s,z (n) ~ 1 (1) n-oo and (2) n-+oo p(p+l) Indeed, let n = , where p is an odd prime. Then it is not difficult to see that 2 Sen) = p and Zen) = p. Therefore, implying (1). We note that if the equation S(Z(n» = Z(S(n» has infinitely many solutions, then clearly the lim inf in (1) is 0, otherwise is 1, since
منابع مشابه
On the Pseudo-smarandache Squarefree Function
In this paper we discuss various problems and conjectures concered the pseudo-Smarandache squarefree function.
متن کاملThe Pseudo-smarandache Function
The Pseudo-Smarandache Function is part of number theory. The function comes from the Smarandache Function. The Pseudo-Smarandache Function is represented by Z(n) where n represents any natural number. The value for a given Z(n) is the smallest integer such that 1+2+3+ . . . + Z(n) is divisible by n. Within the Pseudo-Smarandache Function, there are several formulas which make it easier to find...
متن کاملSolution of Two Questions concerning the Divisor Function and the Pseudo- Smarandache Function
In this paper we completely solve two questions concerning the divisor function and the pseudo Smarandache function.
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 کاملSome 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 کامل