On the hybrid mean value of the Smarandache kn-digital sequence and Smarandache function

2010·0.11 MB·

by  Chan Shi

Scientia Magna
Vol. 6 (2010), No. 4, 20-23
On the hybrid mean value of the Smarandache
kn-digital sequence and Smarandache function 1
Chan Shi
Department of Mathematics, Northwest University,
Xi’an, Shaanxi, P.R.China
E-mail: [email protected]
Abstract Themainpurposeofthispaperisusingtheelementarymethodtostudythehybrid
meanvaluepropertiesoftheSmarandachekn-digitalsequenceandSmarandachefunction,and
give an interesting asymptotic formula for it.
Keywords Smarandache kn-digital sequence, Smarandache function, hybrid mean value, as
-ymptotic formula, elementary method.
§1. Introduction
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 2n and 3n digital sequences a(2,n)
and a(3,n) are defined as {a(2,n)} = {12,24,36,48,510,612,714,816,···} and {a(3,n)} =
{13,26,39,412,515,618,721,824,···}.
Recently, Professor Gou Su told me that she studied the hybrid mean value properties of
the Smarandache kn-digital sequence and the divisor sum function σ(n), and proved that the
asymptotic formula
(cid:88) σ(n) 3π2
= ·lnx+O(1)
a(k,n) k·20·ln10
n≤x
holds for all integers 1≤k ≤9.
WhenIreadprofessorGouSu’swork,Ifoundthatthemethodisverynew,andtheresults
are also interesting. This paper as a note of Gou Su’s work, we consider the hybrid mean value
properties of the Smarandache kn-digital sequence and Smarandache function S(n), which is
defined as the smallest positive integer m such that n|m!. That is, S(n)=min{m:n|m!, m∈
N}. Inthispaper,wewillusetheelementaryandanalyticmethodstostudyasimilarproblem,
and prove a new conclusion. That is, we shall prove the following:
Theorem. Let1≤k ≤9,thenforanyrealnumberx>1,wehavetheasymptoticformula
(cid:88) S(n) 3π2
= ·lnlnx+O(1).
a(k,n) k·20
n≤x
1ThispaperissupportedbytheN.S.F.ofP.R.China.
Vol. 6 OnthehybridmeanvalueoftheSmarandachekn-digitalsequenceandSmarandachefunction 21
§2. Proof of the theorem
In this section, we shall use the elementary and combinational methods to complete the
proof of our theorem. First we need following:
Lemma. For any real number x>1, we have
(cid:181) (cid:182)
(cid:88) S(n) π2 x x
= · +O .
n 6 lnx ln2x
n≤x
Proof. For any real number x>2, from [4] we have the asymptotic formula
(cid:181) (cid:182)
(cid:88) π2 x2 x2
S(n)= · +O . (1)
12 lnx ln2x
n≤x
Then from Euler summation formula (see theorem 3.1 of [3]) we can deduce that
(cid:181) (cid:181) (cid:182)(cid:182) (cid:90) (cid:181) (cid:181) (cid:182) (cid:182)
(cid:88) S(n) 1 π2 x2 x2 x π2 t2 t2 1
= · +O + · +O dt
n x 12 lnx ln2x 12 lnt ln2t t2
1<n≤x 1
(cid:181) (cid:182) (cid:90)
π2 x x π2 x 13π2 x 1
= · +O + · + dt
12 lnx ln2x 12 lnx 12 ln2t
(cid:181) (cid:182) 1
π2 x x
= · +O .
6 lnx ln2x
This proves our Lemma.
Now we take k = 2 (or k = 4), then for any real number x > 1, there exists a positive
integer M such that
5·10M ≤x<5·10M+1,
then we can deduce that
1
M = ·lnx+O(1). (2)
ln10
So from the definition of a(2,n) we have
(cid:88) S(n) (cid:88)4 S(n) (cid:88)49 S(n) (cid:88)499 S(n) 5·1(cid:88)0M−1 S(n)
= + + +···+
a(2,n) a(2,n) a(2,n) a(2,n) a(2,n)
1≤n≤x n=1 n=5 n=50 n=5·10M−1
(cid:88) S(n)
+
a(2,n)
5·10M≤n≤x
(cid:88)4 S(n) (cid:88)49 S(n) (cid:88)499 S(n)
= + + +···
n·(10+2) n·(102+2) n·(103+2)
n=1 n=5 n=50
5·1(cid:88)0M−1 S(n) (cid:88) S(n)
+ + (3)
n·(10M+1+2) n·(10M+2+2)
n=5·10M−1 5·10M≤n≤x
22 ChanShi No. 4
and
(cid:88) S(n) (cid:88)2 S(n) (cid:88)24 S(n) (cid:88)249 S(n) 41·1(cid:88)0M−1 S(n)
= + + +···+
a(4,n) a(4,n) a(4,n) a(4,n) a(4,n)
1≤n≤x n=1 n=3 n=25 n=1·10M−1
4
(cid:88) S(n)
+
a(4,n)
1·10M≤n≤x
4
(cid:88)2 S(n) (cid:88)24 S(n) (cid:88)249 S(n)
= + + +···
n·(10+4) n·(102+4) n·(103+4)
n=1 n=3 n=25
14·1(cid:88)0M−1 S(n) (cid:88) S(n)
+ + . (4)
n·(10M +4) n·(10M+1+4)
n=1·10M−1 1·10M≤n≤x
4 4
Then from (2), (3) and Lemma we may immediately deduce
5·1(cid:88)0k−1 S(n) (cid:88) S(n) (cid:88) S(n)
= −
n·(10k+1+2) n·(10k+1+2) n·(10k+1+2)
n=5·10k−1 n≤5·10k−1 n≤5·10k−1
(cid:181) (cid:182)
π2 5·10k−5·10k−1 1 1
= · · +O
6 10k+1+2 ln(5·10k) k2
(cid:181) (cid:182)
3π2 1 1
= · +O (5)
40 k k2
Similarly,
14·(cid:88)10k−1 S(n) (cid:88) S(n) (cid:88) S(n)
= −
n·(10k+4) n·(10k+4) n·(10k+4)
n=1·10k−1 n≤1·10k−1 n≤1·10k−1
4 4 4 (cid:181) (cid:182)
π2 1 ·10k− 1 ·10k−1 1 1
= · 4 4 · +O
6 10k+4 ln(1 ·10k) k2
(cid:181) (cid:182) 4
3π2 1 1
= · +O . (6)
80 k k2
(cid:88)∞ 1
Noting that the identity =π2/6 and the asymptotic formula
n2
n=1
(cid:181) (cid:182)
(cid:88) 1 1
=lnM +γ+O ,
k M
1≤k≤M
where γ is the Euler constant.
Vol. 6 OnthehybridmeanvalueoftheSmarandachekn-digitalsequenceandSmarandachefunction 23
From (2), (3) and (5) we have
(cid:88) S(n) (cid:88)4 S(n) (cid:88)49 S(n) (cid:88)499 S(n) 5·1(cid:88)0M−1 S(n)
= + + +···+
a(2,n) a(2,n) a(2,n) a(2,n) a(2,n)
1≤n≤x n=1 n=5 n=50 n=5·10M−1
(cid:88) S(n)
+
a(2,n)
5·10M≤n≤x
(cid:195) (cid:33)
(cid:88)M 3π2 1 (cid:88)M 1
= · +O
40 k k2
k=1 k=1
3π2
= lnlnx+O(1).
40
Similarly,
(cid:88) S(n) (cid:88)2 S(n) (cid:88)24 S(n) (cid:88)249 S(n) 41·1(cid:88)0M−1 S(n)
= + + +···+
a(4,n) a(4,n) a(4,n) a(4,n) a(4,n)
1≤n≤x n=1 n=3 n=25 n=1·10M−1
4
(cid:88) S(n)
+
a(4,n)
1·10M≤n≤x
4 (cid:195) (cid:33)
(cid:88)M 3π2 1 (cid:88)M 1
= · +O
80 k k2
k=1 k=1
3π2
= lnlnx+O(1).
80
For using the same methods, we can also prove that the theorem holds for all integers
k =1,3,5,6,7,8,9. This completes the proof of our theorem.
References
[1] F. Smarandache, Sequences of Numbers Involved in Unsolved Problems, Hexis, 2006.
[2] Gou Su, Smarandache 3n-digital sequence and it’s asymptotic properties, Journal of
Inner Mongolia Normal University (Natural Science Edition), 6(2010), No. 39, 450-453.
[3]TomM.Apostol.,IntroductiontoAnalyticalNumberTheory,Spring-Verlag,NewYork,
1976.
[4] Chen Guohui, New Progress on Smarandache Problems, High American Press, 2007.
[5] Kenichiro Kashihara, Comments and topics on Smarandache notions and problems,
Erhus University Press, USA, 1996.
[6]ZhangWenpeng, Theelementarynumbertheory(inChinese), ShaanxiNormalUniver-
sity Press, Xi’an, 2007.
[7]WuNan,OntheSmarandache3n-digitalsequenceandtheZhangWenpeng’sconjecture,
Scientia Magna, 4(2008), No. 4, 120-122.
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Xi’an, 2010.