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Hong, Shaofang Luo, Yuanyuan Qian, Guoyou and Wang, Chunlin 2013. Uniform lower bound for the least common multiple of a polynomial sequence. Comptes Rendus Mathematique, Vol. 351, Issue. 2122, p. 781.
Oon, SheaMing 2013. Note on the Lower Bound of Least Common Multiple. Abstract and Applied Analysis, Vol. 2013, p. 1.
Qian, Guoyou Tan, Qianrong and Hong, Shaofang 2013. On the periodicity of a class of arithmetic functions associated with multiplicative functions. Journal of Number Theory, Vol. 133, Issue. 6, p. 2005.
Qian, Guoyou and Hong, Shaofang 2013. Asymptotic behavior of the least common multiple of consecutive arithmetic progression terms. Archiv der Mathematik, Vol. 100, Issue. 4, p. 337.
Wu, Rongjun Tan, Qianrong and Hong, Shaofang 2013. New lower bounds for the least common multiples of arithmetic progressions. Chinese Annals of Mathematics, Series B, Vol. 34, Issue. 6, p. 861.
Hong, Shaofang Qian, Guoyou and Tan, Qianrong 2012. The least common multiple of a sequence of products of linear polynomials. Acta Mathematica Hungarica, Vol. 135, Issue. 12, p. 160.
QIAN, GUOYOU TAN, QIANRONG and HONG, SHAOFANG 2012. THE LEAST COMMON MULTIPLE OF CONSECUTIVE TERMS IN A QUADRATIC PROGRESSION. Bulletin of the Australian Mathematical Society, Vol. 86, Issue. 03, p. 389.
Let k ≥ 0, a ≥ 1 and b ≥ 0 be integers. We define the arithmetic function gk,a,b for any positive integer n by
If we let a = 1 and b = 0, then gk,a,b becomes the arithmetic function that was previously introduced by Farhi. Farhi proved that gk,1,0 is periodic and that k! is a period. Hong and Yang improved Farhi's period k! to lcm(1, 2, … , k) and conjectured that (lcm(1, 2, … , k, k + 1))/(k + 1) divides the smallest period of gk,1,0. Recently, Farhi and Kane proved this conjecture and determined the smallest period of gk,1,0. For the general integers a ≥ 1 and b ≥ 0, it is natural to ask the following interesting question: is gk,a,b periodic? If so, what is the smallest period of gk,a,b? We first show that the arithmetic function gk,a,b is periodic. Subsequently, we provide detailed padic analysis of the periodic function gk,a,b. Finally, we determine the smallest period of gk,a,b. Our result extends the Farhi–Kane Theorem from the set of positive integers to general arithmetic progressions.
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