An inequality of the Smarandache function

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Scientia Magna Vol. 4 (2008), No. 1, 130-133

An inequality of the Smarandache function Weiyi Zhu College of Mathematics, Physics and Information Engineer, Zhejiang Normal University Jianghua, Zhejiang, P.R.China Abstract For any positive integer n, the famous Smarandache function S(n) is defined as the smallest positive integer m such that n|m!. That is, S(n) = min{m : m ∈ N, n|m!}. In an unpublished paper, Dr. Kenichiro Kashihara asked us to solve the following inequalities n n S (xn 1 ) + S (x2 ) + · · · · · · + S (xn ) ≥ nS (x1 ) · S (x1 ) · · · · · · S (xn ) .

In this paper, we using the elementary method to study this problem, and prove that for any integer n ≥ 1, the inequality has infinite group positive integer solutions (x1 , x2 , · · · , xn ). Keywords F.Smarandache function, inequalities, solution, necessary condition.

§1. Introduction and Results For any positive integer n, the famous F.Smarandache function S(n) is defined as the smallest positive integer m such that n | m!. That is, S(n) = min{m : n | m!, n ∈ N }. For example, the first few values of S(n) are S(1) = 1, S(2) = 2, S(3) = 3, S(4) = 4, S(5) = 5, S(6) = 3, S(7) = 7, S(8) = 4, S(9) = 6, S(10) = 5, S(11) = 11, S(12) = 4, · · · · · · . About the elementary properties of S(n), many authors had studied it, and obtained some interesting results, see reference [2], [3], [4] and [5]. For example, Wang Yongxing [3] studied the mean value properties of S(n), and obtained a sharper asymptotic formula about this function: µ 2 ¶ X π 2 x2 x S(n) = +O . 12 ln x ln2 x n≤x Lu Yaming [4] studied the solutions of an equation involving the F.Smarandache function S(n), and proved that for any positive integer k ≥ 2, the equation S(m1 + m2 + · · · + mk ) = S(m1 ) + S(m2 ) + · · · + S(mk ) has infinite group positive integer solutions (m1 , m2 , · · · , mk ). Jozsef Sandor [5] proved for any positive integer k ≥ 2, there exist infinite group positive integers (m1 , m2 , · · · , mk ) satisfied the following inequality: S(m1 + m2 + · · · + mk ) > S(m1 ) + S(m2 ) + · · · + S(mk ). Also, there exist infinite group positive integers (m1 , m2 , · · · , mk ) such that S(m1 + m2 + · · · + mk ) < S(m1 ) + S(m2 ) + · · · + S(mk ).


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An inequality of the Smarandache function by Don Hass - Issuu