A new limit theorem involving the Smarandache LCM sequence

Scientia Magna Vol. 2 (2006), No. 2, 20-23

A new limit theorem involving the Smarandache LCM sequence Xiaowei Pan Department of Mathematics, Northwest University Xi’an, Shaanxi, P.R.China Abstract The main purpose of this paper is using the elementary method to study the LCM Sequence, and give an asymptotic formula about this sequence.

Keywords

Smarandache LCM Sequence, limitation.

§1. Introduction and results For any positive integer n, we define L(n) is the Least Common Multiply (LCM) of the natural number from 1 through n. That is L(n) = [1, 2, · · · , n].

The Smarandache Least Common Multiply Sequence is defined by: SLS −→ L(1), L(2), L(3), · · · , L(n), · · · .

The first few numbers are: 1, 2, 6, 12, 60, 60, 420, 840, 2520, 2520, · · · .

About some simple arithmetical properties of L(n), there are many results in elementary number theory text books. For example, for any positive integers a, b and c, we have [a, b] =

ab abc · (a, b, c) and [a, b, c] = , (a, b) (a, b)(b, c)(c, a)

where (a1 , a2 , · · · , ak ) denotes the Greatest Common Divisor of a1 , a2 , · · · , ak−1 and ak . But about the deeply arithmetical properties of L(n), it seems that none had studied it before, but it is a very important arithmetical function in elementary number theory. The main purpose of this paper is using the elementary methods to study a limit problem involving L(n), and give an interesting limit theorem for it. That is, we shall prove the following: Theorem. For any positive integer n, we have the asymptotic formula  n1



 L(n2 )     Y   p

!!

3

= e + O exp −c

(ln n) 5 1

(ln ln n) 5

p≤n2

where

Y

denotes the production over all prime p ≤ n2 .

p≤n2

From this Theorem we may immediately deduce the following:

,

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A new limit theorem involving the Smarandache LCM sequence

21

Corollary. Under the notations of above, we have  n1



 L(n2 )    lim  Y  n→∞  p

= e,

p≤n2

where L(n2 ) = [1, 2, · · · , n2 ], p is a prime.

§2. Proof of the theorem In this section, we shall complete the proof of this theorem. First we need the following simple Lemma.

Lemma. For x > 0, we have the asymptotic formula !!

3 X −c(ln x) 5 θ(x) = ln p = x + O x exp , 1 (ln ln x) 5 p≤x where c > 0 is a constant,

X

denotes the summation over all prime p ≤ x.

p≤x

Proof. In fact, this is the different form of the famous prime theorem. Its proof can be found in reference [2].

Now we use this Lemma to prove our Theorem.

Let αs 1 α2 L(n2 ) = [1, 2, · · · , n2 ] = pα 1 p2 · · · ps ,

(1)

be the factorization of L(n2 ) into prime powers, then αi = α(pi ) is the highest power of pi in the factorization of 1, 2, 3, · · · , n2 . Since  n1



 L(n2 )     Y   p





    1 L(n2 )  Y 1   = exp  ln Y  = exp  ln L(n2 ) − ln p  , n n p p≤n2

p≤n2

p≤n2

while ln L(n2 ) − ln

Y

p

=

αn 1 α2 ln (pα 1 p2 · · · pn ) − ln

p≤n2

Y

p

p≤n2

=

X

α(p) ln p −

p≤n2

=

X

X

ln p

p≤n2

(α(p) − 1) ln p

p≤n2

=

X 2 p≤n 3

+

X

(α(p) − 1) ln p +

X n<p≤n2

2 n3

(α(p) − 1) ln p.

(α(p) − 1) ln p

<p≤n

(2)

22

Xiaowei Pan

No. 2 2

In (1), it is clear that if n < pi ≤ n2 , then α(pi ) = 1. If n 3 < pi ≤ n, we have α(pi ) = 2. (In 2 fact if α(pi ) ≥ 3, then p3i > n. This contradiction with pi ≤ n). If pi ≤ n 3 , then α(pi ) ≥ 3. So from these and above Lemma we have X X X (α(p) − 1) ln p = (2 − 1) ln p = ln p, (3) 2

2

2

n 3 <p≤n

n 3 <p≤n

X

X

(α(p) − 1) ln p =

n<p≤n2

n 3 <p≤n

(1 − 1) ln p = 0,

(4)

n<p≤n2





2 X  n3  (α(p) − 1) ln p = O ln2 n 1 = O ln2 n ln n 2 2

!

X

2 = O n 3 ln n .

p≤n 3

p≤n 3

Now combining (2), (3), (4) and (5) we may immediately get 2

Y X p = O n 3 ln n + ln p ln L(n2 ) − ln p≤n2

2

n 3 <p≤n

2

= O n 3 ln n +

X

X

ln p −

ln p

2

p≤n

p≤n 3 3

−c(ln n) 5

2 3

= O n ln n + n + O n exp

1

2

2

2

−n 3 − O n 3 exp

= n + O n exp

(ln ln n) 5 !!

3

−c(ln n 3 ) 5 2 3

1

(ln ln n ) 5 !!

3 −c(ln n) 5 .

1 (ln ln n) 5

That is,  n1



 L(n2 )     Y   p

 =

exp 

 1 ln L(n2 ) − ln n

 Y

p

p≤n2

p≤n2

" !!## 3 −c(ln n) 5 1 n + O n exp exp 1 n (ln ln n) 5 " !!# 3 −c(ln n) 5 exp 1 + O exp 1 (ln ln n) 5 " !!# 3 −c(ln n) 5 e · exp O exp 1 (ln ln n) 5 " !!# 3 −c(ln n) 5 e 1 + O exp 1 (ln ln n) 5 !!

3 −c(ln n) 5 e + O exp .

1 (ln ln n) 5 "

=

=

=

=

=

!!

(5)

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A new limit theorem involving the Smarandache LCM sequence

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This completes the proof of Theorem.

The Corollary follows from Theorem with n â&#x2020;&#x2019; â&#x2C6;&#x17E;.

References [1] Amarnth Murthy, Generalized Partitions and New Ideas On Number Theory and Smarandache Sequences, Hexis, 2005, pp. 20-22.

[2] Pan Chengdong and Pan Chengbiao, The Foundation of Analytic number Theory, Science Publication, 1999, pp. 204-205.

[3] Tom M.Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976.