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6 results

  • AN IMPROVEMENT TO A THEOREM OF LEONETTI AND LUCA

  • Part of
  • TRAN NGUYEN THANH DANH, HOANG TUAN DUNG, PHAM VIET HUNG, NGUYEN DINH KIEN, NGUYEN AN THINH, KHUC DINH TOAN, NGUYEN XUAN THO
  • Journal:
    Bulletin of the Australian Mathematical Society / Volume 109 / Issue 3 / June 2024
    Published online by Cambridge University Press:
    01 September 2023, pp. 437-442
    Print publication:
    June 2024
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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 $X=x_1+x_2+2k$.