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.

Let (a, b) ∈ Z 2 , where b = 0 and (a, b) = (±2, −1). We prove that then there exist two positive relatively prime composite integers x 1 , x 2 such that the sequence given by x n+1 = ax n + bx n−1 , n = 2, 3, . . . , consists of composite terms only, i.e., |x n | is a composite integer for each n ∈ N. In the proof of this result we use certain covering systems, divisibility sequences and, for some special pairs (a, ±1), computer calculations. The paper is motivated by a result of Graham who proved this theorem in the special case of the Fibonacci-like sequence, where (a, b) = (1, 1).

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