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THE LEAST COMMON MULTIPLE OF CONSECUTIVE TERMS IN A QUADRATIC PROGRESSION

Part of: Sequences and sets Elementary number theory

Published online by Cambridge University Press:  12 April 2012

GUOYOU QIAN ,
QIANRONG TAN  and
SHAOFANG HONG
GUOYOU QIAN
Affiliation:
Mathematical College, Sichuan University, Chengdu 610064, PR China (email: qiangy1230@gmail.com, qiangy1230@163.com)
QIANRONG TAN
Affiliation:
School of Computer Science and Technology, Panzhihua University, Panzhihua 617000, PR China (email: tqrmei6@126.com)
SHAOFANG HONG*
Affiliation:
Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, PR China (email: sfhong@scu.edu.cn, s-f.hong@tom.com, hongsf02@yahoo.com)
*
✉For correspondence; e-mail: sfhong@scu.edu.cn
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Abstract

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Let k be any given positive integer. We define the arithmetic function gk for any positive integer n by We first show that gk is periodic. Subsequently, we provide a detailed local analysis of the periodic function gk, and determine its smallest period. We also obtain an asymptotic formula for log lcm0≤i≤k {(n+i)2+1}.

Keywords

quadratic progressionleast common multiplep-adic valuationarithmetic functionsmallest period

MSC classification

Secondary: 11B25: Arithmetic progressions 11A05: Multiplicative structure; Euclidean algorithm; greatest common divisors
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 86 , Issue 3 , December 2012 , pp. 389 - 404
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

Footnotes

S. Hong was supported partially by National Science Foundation of China Grant #10971145 and by the PhD Programs Foundation of Ministry of Education of China Grant #20100181110073.

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