Talk:Smooth number

Powersmooth example

The powersmooth example is correct, though the number is also 16-powersmooth, which is more useful. 2^4 <= 16 I'm new to wikipedia or else I might just change it. User:Erg255

Thanks, yes. I changed it. --Nuffle 14:33, 9 November 2005 (UTC)

Formula for 3-smooth sequence

Is there a formula for the n th number in the 3-smooth sequence? User:SurrealWarrior

Not that I know of. A003586 has an asymptotic formula, though. CRGreathouse (t | c) 19:19, 12 September 2007 (UTC)

Is the distribution formula correct?

The distribution is given as approximately

$\Psi (x,B)\sim {\frac {1}{\pi (B)!}}\prod _{p\leq B}{\frac {\log x}{\log p}}$ .

which is equivalent to

$\Psi (x,B)\sim \prod _{p_{i}\leq B}{\frac {\log x}{i\log p_{i}}}$ .

(counting i from 1)

Each term in the product marks how much adding that term increases the number of B-smooth numbers below x. But the terms can fall below one, which means that the estimate for $\Psi (x,B)$ can go down as B increases, which makes no sense.

Can anyone explain? — ciphergoth 17:48, 17 February 2007 (UTC)

That asymptotic formula is only true for B small (relative to x). Only large values of B cause trouble there. CRGreathouse (t | c) 19:18, 12 September 2007 (UTC)

Unclear meaning

The article is not clear about whether numbers with gaps in their factorisation are considered smooth or not. For example, is 10 5-smooth? Or is 12 5-smooth, since all of it's prime factors are lower than 5? As far as I'm aware, these two examples are untrue, but I have never formally learnt about the topic. N4m3 (talk) 22:48, 10 September 2011 (UTC)

To answer my own question, OEIS seems to show both of my examples as true. I may edit the page. N4m3 (talk) 22:52, 10 September 2011 (UTC)

Terminology

The use of the words smooth/powersmooth against friable/ultrafriable in analytic number theory appeals some comments. This is important inasmuch words of science are part of science.

Almost every one, in or out the subject, agrees that smooth is overused and that friable is better: (a) it is only employed in this context; (b) it is evocative; (c) it is the same in English and in French; (d) it is easy to decline in expression such friability parameter, friable summation, etc.; (e) so-called smooth integers are anything but smooth, since their distribution is rather violently irregular, etc.

Moreover an increasing number of English-writing authors have now chosen to use friable: among others Balog, Blomer, Moree, Luca, De Koninck, Kowalski, Fouvry, Loomis, Banks, Shparlinski, and French mathematicians such as La Bretèche, Tenenbaum and Wu (all specialists of the field) when they write in English. Several other researchers have explained in private discussion that they firmly dislike the use of smooth in this context.

Thus it is simply a matter of will to contribute spreading a better word and help clarifying the literature. In a recent mathematical text dealing with friable integers, one can read: "the idea is now [...] to smoothen the sum". What should the meaning of this word be in such a paper?

We, mathematicians are responsible for our practice, and constitute a small community around the world; surely we can decide what are the best words for us to think about our own concepts and express ourselves in the most effective way. And it is never too late to do things right. Of course all this applies equally well (and even more) in the case of ultrafriable: indeed powersmooth is misleading, one may think that only small exponents are allowed. — Preceding unsigned comment added by Gralaad (talk • contribs) 15:01, 14 January 2014 (UTC) Gralaad (talk) 16:57, 14 January 2014 (UTC)

Such a thing as an "almost smooth" number?

By "almost smooth" I mean that a number is (B,k)-almost-smooth if no more than k of it's factors are greater than B. Jimw338 (talk) 15:02, 24 June 2017 (UTC)