We prove that the sequence [ξ(5/4) n ], n=1,2, . . . , where ξ 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 ([ξ(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 ξ(5/3) n and to ξ(7/5) n , n∈ℕ. The corresponding sets of possible divisors are also described.
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