On the mean value of SSMP(n) and SIMP(n)

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Scientia Magna Vol. 5 (2009), No. 3, 1-5

On the mean value of SSM P (n) and SIM P (n)1 Yiren Wang Department of Mathematics, Northwest University, Xi’an, Shaanxi, P.R.China Abstract The main purpose of this paper it to studied the mean value properties of the Smarandache Superior m-th power part sequence SSM P (n) and the Smarandache Inferior m-th power part sequence SIM P (n), and give several interesting asymptotic formula for them. Keywords Smarandache Superior m-th power part sequence, Smarandache Inferior m-th power part sequences, mean value, asymptotic formula.

§1. Introduction and Results For any positive integer n, the Smarandache Superior m-th power part sequence SSM P (n) is defined as the smallest m-th power greater than or equal to n. The Smarandache Inferior m-th power part sequence SIM P (n) is defined as the largest m-th power less than or equal to n. For example, if m = 2, then the first few terms of SIM P (n) are: 0, 1, 1, 1, 4, 4, 4, 4, 4, 9, 9, 9, 9, 9, 9, 9, 16, 16, 16, 16, 16, 16, 16, 16, 16, 25, · · · . The first few terms of SSM P (n) are: 1, 4, 4, 4, 9, 9, 9, 9, 9, 16, 16, 16, 16, 16, 16, 16, 25, · · · . If m = 3, then The first few terms of SSM P (n) are: 1, 8, 8, 8, 8, 8, 8, 8, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 64, · · · . The first few terms of SIM P (n) are: 0, 1, 1, 1, 1, 1, 1, 1, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 27, · · · . Now we let Sn = (SSM P (1) + SSM P (2) + · · · + SSM P (n))/n; In = (SIM P (1) + SIM P (2) + · · · + SIM P (n))/n; p Kn = n SSM P (1) + SSM P (2) + · · · + SSM P (n); p In = n SIM P (1) + SIM P (2) + · · · + SIM P (n). In reference [2], Dr. K.Kashihara asked us to study the properties of these sequences. Gou Su [3] studied these problem, and proved the following conclusion: For any real number x > 2 and integer m = 2, we have the asymptotic formula X n6x

SSSP (n) =

³ 3´ x2 + O x2 , 2

and Sn In 1 This

X

SISP (n) =

n6x

³ 1´ = 1 + O n− 2 ,

lim Sn n→∞ In

³ 3´ x2 + O x2 , 2

= 1.

work is supported by the Shaanxi Provincial Education Department Foundation 08JK433.


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