×

Counting divisors of Lucas numbers. (English) Zbl 0936.11013

Let \(L_n\) be the sequence of Lucas numbers defined by \(L_0= 2\), \(L_1= 1\) and \(L_n= L_{n-1}+ L_{n-2}\). We say a positive integer \(m\) is a divisor of this sequence if \(m\) divides a Lucas number. The author investigates the density of the set of divisors of the Lucas sequence. The main result of the paper is:
Theorem 1. Let \({\mathcal L}(x)\) denote the number of divisors not exceeding \(x\) of the sequence of Lucas numbers. Then, for \(t\geq 1\), \[ {\mathcal L}(x)= \frac{x}{\log x} \Biggl( \sum_{j=0}^{t-1} c_j\cdot \log^{2^{-j}/3}x+ O(\log^{2^{-t}/3}x) \Biggr), \] where \(c_0,\dots, c_t\) are positive constants and the implied constant depends at most on \(t\).
Reviewer: P.Kiss (Eger)

MSC:

11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11B05 Density, gaps, topology
PDFBibTeX XMLCite
Full Text: DOI