Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1995-1996. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 23)
KlSS, P. and ZAY, B., A note on the prime divisors of Lucas numbers
18 Péter Kiss and Béla Zay The purpose of this note is to study the reciprocal sum of the prime divisors of Lucas numbers. Let /(n) be the reciprocal sum of prime divisors of R n , i.e. let /W=E; ( n >°)p\R n 1 We write f(n ) = 0 if R n = ±1 ( R n ^ 0 for n > 0 since a/ß is not a root of unity). The value of f(n) can be arbitrarily large (e.g. in the case n = m\ f(n ) > log log log n) and arbitrarily small (e.g. if n is a prime then f[n) < (log log n) 2/ ny But the average order of f{n ) is a constant f(n) = c 0x + O(log log x) n<.x (see P. Kiss [3]). It is proved in [2] for the special case (A; B) = (3;—2), that if R n = 2 n — 1 is the sequence of Mersenne numbers, then the average order of f(n ) is also a constant even if we consider the function in a short interval. For general sequences we show a similar result. Theorem. Let x and z be positive integers such that z -— > oo as x — > oo. log log x Then for any sufficiently large x we have x+z f( n) = cz + o(z), n=x where c > 0 is a constant depending only on the parameters of the sequence. For the proof of theorem we need some auxiliary results. In the proofs Ci,C2, ... will denote positive constans depending only on the sequence. Furthermore we shall use some elementary results of prime number theory, they can be found e.g. in [1]. Lemma 1. For the reciprocal sum of the primes for which r(p) < y we have V -= log logy +0(1) Í—/ p r(p)<y for any sufficiently large positive y.
