We prove that the sequence [xi(5/4)(n)], n= 1,2,..., where xi is an arbitrary positive number, contains infinitely many composite numbers. A corresponding result for the sequences [(3/2)(n)] and [(4/3)(n)], n= 1,2,..., was obtained by Forman and Shapiro in 1967. Furthermore, it is shown that there are infinitely many positive integers n such that ([xi(5/4)(n)], 6006)> 1, where 6006= 2.3.7.11.13. Similar results are obtained for shifted powers of some other rational numbers. In particular, the same is proved for the sets of integers nearest to xi(5/3)(n) and to xi(7/5)(n), n is an element of N. The corresponding sets of possible divisors are also described.
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