On the Sandor-Smarandache Function

Abstract

In the current literature, a new Smarandache-type arithmetic function, involving binomial coefficients, has been proposed by Sandor. The new function, denoted by SS(n), is named the Sandor-Smarandache function. It has been found that, like many Smarandache-type functions, SS(n) is not multiplicative. Sandor found SS(n) when n (≥3) is an odd integer. Since then, the determination of SS(n) for even n remains a challenging problem. It has been shown that the function has a simple form even when n is even and not divisible by 3. This paper finds SS(n) in some particular cases of n, and finds an upper bound of SS(n) for some special forms of n. Some equations involving the Sandor-Smarandache function and pseudo-Smarandache function have been studied. A list of values of SS(n) for n = 1(1)480, calculated on a computer, is appended at the end of the paper.

A Unified Approach to the Sandor-Smarandache Function

“…Islam et al [3] proved the results below. Lemma 1.1: SS(n) = n -2 if and only if n (  7) is an odd integer.…”

“…Later, the problem was studied to some extent by Islam, and Majumdar [6], who derived the expressions of SS(2mp), SS(6mp), SS(60mp), and SS(420mp), where p is an odd prime and m is any (positive) integer. Islam et al [3] subsequently found explicit forms of SS(6t), SS(12t), SS(18t), SS(42t), SS(30t), and SS(210t) for some particular cases of t. Later, Majumdar and Ahmed [7] extended the results of Islam et al [3] by considering all the possible cases involved in SS(210t). Recently, Islam and Majumdar [4] derived the expressions of SS(120m), SS(840m), SS(9240m), and SS(120120m) for some particular cases of m.…”

“…These are given in the following lemmas. For proof, the readers are referred to Islam et al [3]. Lemma 2.1: (Fundamental Theorem of Arithmetic) Let a and b be two (positive) integers with (a, b) = 1.…”

Palindromes in Some Smarandache-Type Functions

“…Some results related to the Sandor-Smarandache function are given in Majumdar [6]. The function was later studied in more detail by Islam and Majumdar [12], Islam, Gunarto and Majumdar [13], and more recently by [14] and [15]. The following result is also known about the Sandor-Smarandache function.…”