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Leonetti and Luca ['On the iterates of the shifted Euler's function', Bull. Aust. Math. Soc. , to appear] have shown that the integer sequence $(x_n)_{n\geq 1}$ defined by $x_{n+2}=\phi (x_{n+1})+\phi (x_{n})+k$ , where $x_1,x_2\geq 1$ , $k\geq 0$ and $2 \mid k$ , is bounded by $4^{X^{3^{k+1}}}$ , where $X=(3x_1+5x_2+7k)/2$. We improve this result by showing that the sequence $(x_n)$ is bounded by $2^{2X^2+X-3}$ , where $X=x_1+x_2+2k$. [ABSTRACT FROM AUTHOR]

We study a recursively defined sequence which is constructed using the least common multiple. Several authors have conjectured that every term of that sequence is or a prime. In this paper we show that this claim is connected to a strong version of Linnik's theorem, which is still unproved. We also study a generalization that replaces the first term by any positive integer. Under this variation some composite numbers may appear now. We give a full characterization of these numbers. [ABSTRACT FROM AUTHOR]