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We consider a version of the forest fire model on graph $G$, where eachvertex of a graph becomes occupied with rate one. A fixed vertex $v_0$ is hitby lightning with the same rate, and when this occurs, the whole cluster ofoccupied vertices containing $v_0$ is burnt out. We show that when $G=Z_{+}$,the times between consecutive burnouts at vertex $n$, divided by $\log n$,converge weakly as $n\to\infty$ to a random variable which distribution is$1-\rho(x)$ where $\rho(x)$ is the Dickman function.  We also show that on transitive graphs with a non-trivial site percolationthreshold and one infinite cluster at most, the distributions of the time tillthe first burnout of {\it any} vertex have exponential tails.  Finally, we give an elementary proof of an interesting limit:$\lim_{n\to\infty} \sum_{k=1}^n {n \choose k} (-1)^k \log k -\log\logn=\gamma$.

Forest fires on $\Z_+$ with ignition only at 0