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AN IMPROVEMENT TO A THEOREM OF LEONETTI AND LUCA
- Part of
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- 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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- Article
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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 
$X=x_1+x_2+2k$.
