Scientia Magna Vol. 8 (2012), No. 1, 31-36
On the hybrid mean value of the Smarandache kn digital sequence with SL(n) function and divisor function d(n)1 Le Huan Department of Mathematics, Northwest University, Xi’an, Shaanxi, P. R. China E-mail: huanle1017@163.com Abstract The main purpose of this paper is using the elementary method to study the hybrid mean value properties of the Smarandache kn digital sequence with SL(n) function and divisor function d(n), then give two interesting asymptotic formulae for it. Keywords Smarandache kn digital sequence, SL(n) function, divisor function, hybrid mean value, asymptotic formula.
§1. Introduction and results For any positive integer k, the famous Smarandache kn-digital sequence a(k, n) is defined as all positive integers which can be partitioned into two groups such that the second part is k times bigger than the first. For example, Smarandache 3n digital sequences a(3, n) is defined as {a(3, n)} = {13, 26, 39, 412, 515, 618, 721, 824, · · · }, for example, a(3, 15) = 1545. In the reference [1], Professor F. Smarandache asked us to study the properties of a(k, n), about this problem, many people have studied and obtained many meaningful results. In [2], Lu Xiaoping studied the mean value of this sequence and gave the following theorem: X n≤N
9 n = · ln N + O(1). a(5, n) 50 ln 10
In [3], Gou Su studied the hybrid mean value of Smarandache kn sequence and divisor function σ(n), and gave the following theorem: X σ(n) 3π 2 = · ln x + O(1), a(k, n) k · 20 · ln 10
n≤x
where 1 ≤ k ≤ 9. Inspired by the above conclusions, in this paper, we study the hybrid mean value properties of the Smarandache kn-digital sequence with SL(n) function and divisor function d(n), where 1 This work is supported by Scientific Research Program Funded by Shaanxi Provincial Education Department(No. 11JK0470).
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SL(n) is defined as the smallest positive integer k such that n|[1, 2, . . . , k], that is SL(n) = min{k : k ∈ N, n|[1, 2, . . . , k]}. And obtained the following results： Theorem 1.1. Let 1 ≤ k ≤ 9, then for any real number x > 1, we have the asymptotic formula X SL(n) 3π 2 = · ln ln x + O(1). a(k, n) k · 20
n≤x
Theorem 1.2. Let 1 ≤ k ≤ 9, then for any real number x > 1, we have the asymptotic formula X d(n) · SL(n) π4 = · ln ln x + O(1). a(k, n) k · 20
n≤x
§2. Lemmas Lemma 2.1. For any real number x > 1, we have µ ¶ X SL(n) π2 x x = · +O . n 6 ln x ln2 x
n≤x
Proof. For any real number x > 1, by reference [4] we have the asymptotic formula X n≤x
π 2 x2 SL(n) = · +O 12 ln x
µ
x2 ln2 x
¶ .
Using Abel formula (see [6]) we get X SL(n) n
1 x
=
1<n≤x
µ
π2 12 π2 6
= =
µ 2 ¶¶ Z x µ 2 µ 2 ¶¶ π 2 x2 x t 1 π t2 · +O + · + O dt 2 2 12 ln x 12 ln t ln x ln2 t 1 t µ ¶ µZ x ¶ Z x x π2 x 1 1 · +O + dt + O dt 2 ln x 12 1 ln t ln2 x 1 ln t µ ¶ x x · +O . ln x ln2 x
This proves Lemma 2.1. Lemma 2.2. For any real number x > 1, we have µ ¶ X d(n) · SL(n) π4 x x = · +O . n 18 ln x ln2 x
n≤x
Proof. For any real number x > 1, by reference [5] we have the asymptotic formula X n≤x
d(n) · SL(n) =
π 4 x2 · +O 36 ln x
µ
x2 ln2 x
¶ .
On the hybrid mean value of the Smarandache kn digital sequence with SL(n) function and divisor function d(n)
Vol. 8
33
Using Abel formula (see [6]) we get X d(n) · SL(n) n
=
1<n≤x
= =
1 x
µ 2 ¶¶ Z x µ 4 µ 2 ¶¶ π 4 x2 x 1 π t2 t · +O + · +O dt 2 2 36 ln x t 36 ln t ln x ln2 t 1 µ ¶ µZ x ¶ Z x x π4 x 1 1 · +O + dt + O 2 dt ln x 36 1 ln t ln2 x 1 ln t µ ¶ x x · +O . ln x ln2 x
µ
π4 36 π4 18
This proves Lemma 2.2.
§3. Proof of the theorems In this section, we shall use the elementary and combinational methods to complete the proof of our theorems. We just prove the case of k = 3 and k = 5, for other positive integers we can use the similar methods. First we prove theorem 1.1. Let k = 3, for any positive integer x > 3, there exists a positive integer M such that · · 33} < x ≤ 33 · · 33} . |33 ·{z | ·{z M
M +1
So 10M − 1 < 3x ≤ 10M +1 − 1, Then we have ln 3x −1−O ln 10
µ
1 10M
¶
ln 3x ≤M < −O ln 10
µ
1 10M
¶ .
(1)
By the definition of a(3, n) we have X SL(n) a(3, n)
1≤n≤x
3 33 333 X X SL(n) X SL(n) SL(n) = + + + ··· + a(3, n) n=4 a(3, n) n=34 a(3, n) n=1
+
X 1 M 3 ·10 ≤n≤x
=
M 1 3 ·10 −1
X
n= 13 ·10M −1
SL(n) a(3, n)
SL(n) a(3, n)
3 X
33 333 X X SL(n) SL(n) SL(n) + + + ··· 2 n(10 + 3) n=4 n(10 + 3) n=34 n(103 + 3) n=1 M 1 3 ·10 −1
+
X
n= 13 ·10M −1
SL(n) + n(10M +1 + 3)
X 1 M 3 ·10 ≤n≤x
SL(n) . n(10M +2 + 3)
(2)
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Form (1), (2) and lemma 2.1 we get k 1 3 ·10 −1
X
n= 13 ·10k−1
SL(n) n · (10k + 3)
X
=
1 k 3 ·10 −1
X n= 31 ·10k−1
SL(n) n · (10k + 3)
· 10k − 13 · 10k−1 1 · +O 10k + 3 ln( 13 · 10k ) µ ¶ 3π 2 1 1 . · +O 3 · 20 k k2 π2 · 6
= =
Note that the identity
SL(n) − n · (10k + 3)
1 3
µ
1 k2
¶
(3)
∞ X 1 = π 2 /6 and the asymptotic formula 2 n n=1
X 1≤k≤M
1 = ln M + γ + O k
µ
1 M
¶ ,
where γ is Euler’s constant. Form (1), (2) and (3) we get X SL(n) a(3, n)
1≤n≤x
3 33 333 X X SL(n) X SL(n) SL(n) = + + + ··· + a(3, n) a(3, n) a(4, n) n=1 n=4 n=34
n= 13 ·10M −1
SL(n) a(3, n)
X
k=1
=
X
SL(n) a(3, n) 1 M 3 ·10 ≤n≤x ! ÃM M X X 1 3π 2 1 · +O 3 · 20 k k2 +
=
M 1 3 ·10 −1
k=1
3π 2 ln ln x + O(1). 3 · 20
Now we prove the case of k = 5, for any positive integer x > 1, there exists a positive integer M such that · · · 99} . · · · 00} < x ≤ 199 | {z |200 {z M
M +1
So 10M < 5x ≤ 10M +1 − 5, Then we have ln 5x −1−O ln 10
µ
1 10M
¶ ≤M <
ln 5x . ln 10
(4)
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On the hybrid mean value of the Smarandache kn digital sequence with SL(n) function and divisor function d(n)
35
By the definition of a(5, n) we have X SL(n) a(5, n)
1≤n≤x
19 199 X SL(n) X X SL(n) SL(n) = + + + ··· + a(5, n) n=2 a(5, n) n=20 a(5, n) n=1
X
+
1 M 5 ·10 ≤n≤x
=
M 1 5 ·10 −1
X
n= 15 ·10M −1
SL(n) a(5, n)
SL(n) a(5, n)
19 199 X X SL(n) SL(n) SL(n) + + + ··· 2 n(10 + 5) n=2 n(10 + 5) n=20 n(103 + 5) n=1
X
M 1 5 ·10 −1
X
+
n= 15 ·10M −1
SL(n) + n(10M +1 + 5)
X 1 M 5 ·10 ≤n≤x
SL(n) . n(10M +2 + 5)
(5)
Form (4), (5) and lemma 2.1 we get k 1 5 ·10 −1
X
n= 15 ·10k−1
SL(n) n · (10k + 5)
=
X 1 k 5 ·10 −1
2
= =
X SL(n) SL(n) − n · (10k + 5) 1 k−1 n · (10k + 5)
1 5
5 ·10
k
1 5
k−1
· 10 − · 10 10k + 5 µ ¶ 1 3π 2 1 · +O . 5 · 20 k k2 π · 6
1 · +O 1 ln( 5 · 10k )
µ
1 k2
¶
Similar to the proof k = 3, we get X SL(n) a(5, n)
1≤n≤x
19 199 X SL(n) X X SL(n) SL(n) = + + + ··· + a(5, n) a(5, n) a(5, n) n=1 n=2 n=20
X
n= 15 ·10M −1
SL(n) a(5, n)
X
SL(n) a(5, n) 1 M 5 ·10 ≤n≤x ÃM ! M X X 1 3π 2 1 · +O 5 · 20 k k2 +
=
M 1 5 ·10 −1
k=1
k=1
3π 2 ln ln x + O(1). = 5 · 20 By using the same methods, we can also prove that the theorem holds for all integers 1 ≤ k ≤ 9. This completes the proof of theorem 1.1. Similar to the proof of theorem 1.1, we can immediately prove theorem 1.2, we don’t repeated here. As the promotion of this article, we can consider the hybrid mean value of Smarandache kn sequence with other functions such as SL∗ (n), Sdf (n), σ(S(n)), Ω(S ∗ (n)), and obtain the corresponding asymptotic formula.
References [1] F. Smarandache, Sequences of Numbers Involved in Unsolved Problems, American Research Press, 2006.
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[2] Lu Xiaoping, On the Smarandache 5n-digital sequence, 2010, 68-73. [3] Gou Su, Hybrid mean value of Smarandache kn-digital series with divisor sum function, Journal of Xiâ&#x20AC;&#x2122;an Shiyou University, 26 (2011), No. 2, 107-110. [4] Chen Guohui, New Progress on Smarandache Problems, High American Press, 2007. [5] Lv Guoliang, On the hybrid mean value of the F. Smarandache LCM function and the Dirichlet divisor function, Pure and Applied Mathematics, 23 (2007), No. 3, 315-318. [6] Tom M. Apostol, Introduction to Analytical Number Theory, New York, Spring-Verlag, 1976. [7] Zhang Wenpeng, The elementary number theory (in Chinese), Shaanxi Normal University Press, Xiâ&#x20AC;&#x2122;an, 2007.