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Does anyone know of the best estimates for $n_i$ and $n_{i+1}-n_i$ where $n_i$ is the $i-$th $y-$smooth number?

The best I could find was Tijdemann's estimate for the gap in terms of $\frac{n_i}{(log(n_i))^c}$ where $c$ is a constant depending on y.

For $n_i$ itself one could "invert" $\Psi(n_i,y)$ the estimate for number of smooth numbers, and get an estimate in terms if $i,$ but I wonder if it is worked out or if there is a better way.

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  • $\begingroup$ You might find useful to look at Granville's and Friedlander's paper MR1253496 (95b:11086) $\endgroup$
    – Alvin
    May 7, 2013 at 16:29
  • $\begingroup$ Thank you! They do improve Tijdemann's result in the course of that paper, but using abc conjecture. They also mention a result by Stewart and Yu (unconditional) in the same direction. $\endgroup$ May 7, 2013 at 16:48
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    $\begingroup$ The terminology "smooth" is admittedly entrenched in the (English) literature on this topic, but it's a poorly chosen term. The more modern term, which hopefully will oust the older term, is "friable". $\endgroup$ May 7, 2013 at 17:52

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