The main purpose of this paper is using the elementary method to study the
LCM Sequence, and give an asymptotic formula about this sequence

Xiaowei, Pan

English

ZENODO

Scientia MagnaVol. 2 (2006), No. 2, 20-23A new limit theorem involving the SmarandacheLCM sequenceXiaowei PanDepartment of Mathematics, Northwest UniversityXi’an, Shaanxi, P.R.ChinaAbstract The main purpose of this paper is using the elementary method to study theLCM Sequence, and give an asymptotic formula about this sequence.Keywords Smarandache LCM Sequence, limitation.§1. Introduction and resultsFor any positive integer n, we define L(n) is the Least Common Multiply (LCM) of thenatural number from 1 through n. That isL(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 elementarynumber theory text books. For example, for any positive integers a, b and c, we have[a, b] =ab(a, b)and [a, b, c] =abc · (a, b, c)(a, b)(b, c)(c, a),where (a1, a2, · · · , ak) denotes the Greatest Common Divisor of a1, a2, · · · , ak−1 and ak. Butabout the deeply arithmetical properties of L(n), it seems that none had studied it before, butit is a very important arithmetical function in elementary number theory. The main purpose ofthis paper is using the elementary methods to study a limit problem involving L(n), and givean interesting limit theorem for it. That is, we shall prove the following:Theorem. For any positive integer n, we have the asymptotic formulaL(n2)∏p≤n2p1n= e+O(exp(−c (lnn)35(ln lnn)15)),where∏p≤n2denotes the production over all prime p ≤ n2.From this Theorem we may immediately deduce the following:Vol. 2 A new limit theorem involving the Smarandache LCM sequence 21Corollary. Under the notations of above, we havelimn→∞L(n2)∏p≤n2p1n= e,where L(n2) = [1, 2, · · · , n2], p is a prime.§2. Proof of the theoremIn this section, we shall complete the proof of this theorem. First we need the followingsimple Lemma.Lemma. For x > 0, we have the asymptotic formulaθ(x) =∑p≤xln p = x+O(x exp(−c(lnx) 35(ln lnx)15)),where c > 0 is a constant,∑p≤xdenotes the summation over all prime p ≤ x.Proof. In fact, this is the different form of the famous prime theorem. Its proof can befound in reference [2].Now we use this Lemma to prove our Theorem.LetL(n2) = [1, 2, · · · , n2] = pα11 pα22 · · · pαss , (1)be the factorization of L(n2) into prime powers, then αi = α(pi) is the highest power of pi inthe factorization of 1, 2, 3, · · · , n2. SinceL(n2)∏p≤n2p1n= exp 1n ln L(n2)∏p≤n2p = exp 1nlnL(n2)− ln ∏p≤n2p ,whilelnL(n2)− ln∏p≤n2p = ln (pα11 pα22 · · · pαnn )− ln∏p≤n2p=∑p≤n2α(p) ln p−∑p≤n2ln p=∑p≤n2(α(p)− 1) ln p=∑p≤n 23(α(p)− 1) ln p+∑n23<p≤n(α(p)− 1) ln p+∑n<p≤n2(α(p)− 1) ln p. (2)22 Xiaowei Pan No. 2In (1), it is clear that if n < pi ≤ n2, then α(pi) = 1. If n 23 < pi ≤ n, we have α(pi) = 2. (Infact if α(pi) ≥ 3, then p3i > n. This contradiction with pi ≤ n). If pi ≤ n23 , then α(pi) ≥ 3. Sofrom these and above Lemma we have∑n23<p≤n(α(p)− 1) ln p =∑n23<p≤n(2− 1) ln p =∑n23<p≤nln p, (3)∑n<p≤n2(α(p)− 1) ln p =∑n<p≤n2(1− 1) ln p = 0, (4)∑p≤n 23(α(p)− 1) ln p = Oln2 n ∑p≤n 231 = O(ln2 n n 23lnn)= O(n23 lnn). (5)Now combining (2), (3), (4) and (5) we may immediately getlnL(n2)− ln∏p≤n2p = O(n23 lnn)+∑n23<p≤nln p= O(n23 lnn)+∑p≤nln p−∑p≤n 23ln p= O(n23 lnn)+ n+O(n exp(−c(lnn) 35(ln lnn)15))−n 23 −O(n23 exp(−c(lnn 23 ) 35(ln lnn23 )15))= n+O(n exp(−c(lnn) 35(ln lnn)15)).That is,L(n2)∏p≤n2p1n= exp 1nlnL(n2)− ln ∏p≤n2p= exp[1n[n+O(n exp(−c(lnn) 35(ln lnn)15))]]= exp[1 +O(exp(−c(lnn) 35(ln lnn)15))]= e · exp[O(exp(−c(lnn) 35(ln lnn)15))]= e[1 +O(exp(−c(lnn) 35(ln lnn)15))]= e+O(exp(−c(lnn) 35(ln lnn)15)).Vol. 2 A new limit theorem involving the Smarandache LCM sequence 23This completes the proof of Theorem.The Corollary follows from Theorem with n→∞.References[1] Amarnth Murthy, Generalized Partitions and New Ideas On Number Theory andSmarandache 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.

A new limit theorem involving the Smarandache LCM sequence

Abstract

Similar works

Full text

Available Versions

ZENODO

NEUROSURGERY ENTHUSIASTIC WOMEN SOCIETY