Dedicated to the memory of our friend Andrzej Mqkowski On the Distribution of Numbers n Satisfying the Congruence 2(n-k)equivalent to 1 (mod n) for k= 2 and k= 4

In this paper we address an old problem concerning the existence of infinitely many solutions n of the congruence 2(n-k) equivalent to 1 (mod n) for an arbitrary positive integer k. The existence of infinitely many solutions of that congruence follows from more general but not constructive theorems, which do not give an answer about the number of solutions below a given limit x. It is well known that if k = 1, then our congruence hold for every prime number n > 2 as well as for infinitely many odd composite integers n, called pseudoprimes. If k = 3 then every number n of the form 3p (p an odd prime) is a solution of the congruence 2(n-3) (mod n). We study the distribution of consecutive solutions of our congruence in the two simplest but resistant cases k = 2 and k = 4