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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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You might find useful to look at Granville's and Friedlander's paper MR1253496 (95b:11086) –  Alvin May 7 '13 at 16:29
    
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. –  Sankar Sitaraman May 7 '13 at 16:48
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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". –  Greg Martin May 7 '13 at 17:52

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