An infinity series involving the Smarandache-type function

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Scientia Magna Vol. 3 (2007), No. 3, 52-55

An infinity series involving the Smarandache-type function Jing Li Department of Mathematics, Northwest University Xi’an, Shaanxi, P.R.China lj5118@163.com Abstract In this paper, we using the elementary method to study the convergent property of one class Dirichlet series involving a special sequences, and give several interesting identities for it. Keywords Riemann zeta-function, infinity series, identity.

§1. Introduction and Results For any positive integer n and m ≥ 2, the Smarandache-type function Bm (n) is defined as the largest m-th power dividing n. That is, Bm (n) = max{xm : xm | n}(∀n ∈ N ∗ ). For example, B2 (1) = 1, B2 (2) = 1, B2 (3) = 1, B2 (4) = 2, B2 (5) = 1, B2 (6) = 1, B2 (7) = 1, B2 (8) = 2, B2 (9) = 3, · · · . This function was first introduced by Professor Smarandache. In [1], Henry Bottomley presented that Bm (n) is a multiplicative function. That is, (∀a, b ∈ N )(a, b) = 1 ⇒ Bm (a · b) = Bm (a) · Bm (b). It is easily to show that Bm (pα ) = pim , α = im + l, (i ≥ 0, 0 ≤ l < m), where p is a prime. αr 1 α2 So, if n = pα 1 p2 · · · pr is the prime powers decomposition of n, then the following identity is obviously: i1 m i2 m αr 1 α2 Bm (n) = Bm (pα · · · pirr m , 1 p2 · · · pr ) = p 1 p2 where αj = ij m + lj (ij ≥ 0, 0 ≤ lj < m). Similarly, for any positive integer n and any fixed positive integer m, we define another Smarandache function Cm (n) as following: Cm (n) = max{x ∈ N : xm | n}(∀n ∈ N ∗ ). Obviously, Cm (n) is also a multiplicative function. From the definition of Bm (n) and Cm (n), we may immediately get Bm (n) = Cm (n)m .


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