An upper bound for the moments of a GCD related to Lucas sequences. (English) Zbl 1418.11027

Summary: Let \((u_n)_{n \geq 0}\) be a non-degenerate Lucas sequence, given by the relation \(u_n=a_1 u_{n-1}+a_2 u_{n-2}\). Let \(\ell_u(m)=\mathrm{lcm}(m, z_u(m))\), for \((m,a_2)=1\), where \(z_u(m)\) is the rank of appearance of \(m\) in \(u_n\). We prove that \[ \sum_{\substack{m>x\\ (m,a_2)=1}}\frac{1}{\ell_u(m)}\leq \exp \biggl(-\biggl(\frac 1{\sqrt{6}}-\varepsilon +o(1)\biggr)\sqrt{(\log x)(\log \log x)}\biggr),\] when \(x\) is sufficiently large in terms of \(\varepsilon\), and where the \(o(1)\) depends on \(u\). Moreover, if \(g_u(n)=\gcd (n,u_n)\), we show that for every \(k\geq 1\), \[\sum_{n\leq x}g_u(n)^{k}\leq x^{k+1}\exp (-(1+o(1)) \sqrt{(\log x)(\log \log x)}),\] when \(x\) is sufficiently large, and where the \(o(1)\) depends upon \(u\) and \(k\). This gives a partial answer to a question posed by C. Sanna. As a by-product, we derive bounds on \(\#\{n\leq x: (n, u_n)>y\}\), at least in certain ranges of \(y\), which strengthens what was already obtained by Sanna. Finally, we begin the study of the multiplicative analogs of \(\ell _u(m)\), finding interesting results.


11B37 Recurrences
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors
11N64 Other results on the distribution of values or the characterization of arithmetic functions
Full Text: DOI arXiv Euclid


[1] J.J. Alba González, F. Luca, C. Pomerance and I.E. Shparlinski, On numbers \(n\) dividing the \(n\)th term of a linear recurrence, Proc. Edinb. Math. Soc. 55 (2012), 271-289. · Zbl 1262.11015
[2] R. André-Jeannin, Divisibility of generalized Fibonacci and Lucas numbers by their subscripts, Fibonacci Quart. 29 (1991), 364-366. · Zbl 0737.11003
[3] T. Apostol, Introduction to analytic number theory, Springer, 1976. · Zbl 0335.10001
[4] E.R. Canfield, P. Erdős and C. Pomerance, On a problem of Oppenheim concerning “factorisatio numerorum”, J. Number Theory 17 (1983), 1-28. · Zbl 0513.10043
[5] G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence sequences, Math. Surveys and Monographs 104, Amer. Math. Soc., Providence, RI, 2003. · Zbl 1033.11006
[6] A. Ivić and C. Pomerance, Estimate for certain sums involving the largest prime factor of an integer, pp. 769-789 in Topics in classical number theory, vol. I (Budapest, 1981), Colloq. Math. Soc. János Bolyai 34, North-Holland, Amsterdam, 1984.
[7] P. Leonetti and C. Sanna, On the greatest common divisor of \(n\) and the \(n\)th Fibonacci number, Rocky Mountain J. Math. 48 (2018), 1191-1199. · Zbl 1437.11025
[8] F. Luca and E. Tron, The distribution of self-Fibonacci divisors, pp. 149-158 in Advances in the theory of numbers, Fields Inst. Commun. 77, Fields Inst. Res. Math. Sci., Toronto, 2015.
[9] M. Renault, The period, rank, and order of the \((a, b)\)-Fibonacci sequence mod \(m\), Math. Mag. 86 (2013), 372-380. · Zbl 1293.11025
[10] C. Sanna, The moments of the logarithm of a G.C.D. related to Lucas sequences, J. Number Theory 191 (2018), 305-315. · Zbl 1444.11028
[11] —-, On numbers \(n\) dividing the \(n\)th term of a Lucas sequence, Int. J. Number Theory 13 (2017), 725-734. · Zbl 1416.11029
[12] —-, On numbers \(n\) relatively prime to the \(n\)th term of a linear recurrence, Bull. Malays. Math. Sci. Soc. 42 (2019), 827-833. · Zbl 1458.11028
[13] C. Sanna and E. Tron, The density of numbers \(n\) having a prescribed G.C.D. with the \(n\)th Fibonacci number, Indag. Math. (N.S.) 29 (2018), 972-980. · Zbl 1417.11012
[14] C. Smyth, The terms in Lucas sequences divisible by their indices, J. Integer Seq. 13 (2010), article 10.2.4. · Zbl 1210.11025
[15] L. Somer, Divisibility of terms in Lucas sequences by their subscripts, pp. 515-525 in Applications of Fibonacci numbers, vol. 5 (St. Andrews, 1992), Kluwer Acad. Publ., Dordrecht, 1993.
[16] G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge Stud. Advanced Math. 46, Cambridge Univ. Press, 1995.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.