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Article: On the functional equation [(S(n)).sup.r] + [(S(n)).sup.r-1] + ... + S(n) = n.
- Article from:
- Smarandache Notions Journal
- Article date:
- January 1, 2000
- Author:
CopyrightCOPYRIGHT 2000 American Research Press. This material is published under license from the publisher through the Gale Group, Farmington Hills, Michigan. All inquiries regarding rights should be directed to the Gale Group. (Hide copyright information)
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Abstract For any positive integer n, let S (n) be the Smarandache function of n. Let r be a fixed positive integer with r [greater than or equal to] 3. In this paper we give a necessary and sufficient condition for the functional equation [(S (n)).sup.r] + [(S(n)).sup.r - 1] + ... + S (n) = n to have positive integer solutions n.
Key words Smarandache function, functional equation, solvability.
1 Introduction
Let [??] be the set of all positive integers. For any n [member of] [??], let the arithmetic function
(1) S(n) = min{a|a[member of][??], n|a!}
Then S (n) is called the Smarandache function of n For a fixed r [member of] ...