A generalization of Everett's result on the Collatz \(3x+1\) problem
For the problem of the title \textit{C. J. Everett} [Adv. Math. 25, 42-45 (1977; Zbl 0352.10001)] has proved that almost every natural number m, \(m>1\), has an iterate less than m. The author generalizes this result as follows: Let N be a natural number, x a real number and \(\pi_ k(x)=card\{m\leq x\); m a natural number and for some natural number n, \(m_{n+p}<m\), \(0\leq p\leq k-1\}\). Then \(\pi_ k(x)/x\to 1\) as \(x\to \infty\).
scientific article; zbMATH DE number 4057597