If we wait a few days, Neil, we might get some enlightenment from Rich or from somebody who can read that French math book. And then we'll have a much more substantive comment to add. (For example, suppose it turns out that Tenenbaum and Wu made a mistake in their solution to Exercise 28? That'll be a great example of OEIS-enabled math.) On Fri, Apr 10, 2020 at 6:15 PM Neil Sloane <njasloane@gmail.com> wrote:
Allan,
1. There are two related sequences. If we call your A063539 the number of sqrt(n-1)-smooth numbers , then A295084 counts sqrt(n)-smooth numbers, and my A333534 counts log(n)-smooth numbers. I only added it two days ago - a strange coincidence? I am interested in smooth numbers because they seem to underlie A332580, for which I am trying to find an upper bound.
2. There is a book by Andrew Granville on Smooth Numbers - I found a copy on the web:
http://math.utoledo.edu/~codenth/Spring_13/3200/NT-books/Smooth_Numbers-Comp...
3. Will it be OK if I add your comments about the question of the density of A63539 to that entry?
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Fri, Apr 10, 2020 at 4:59 PM Allan Wechsler <acwacw@gmail.com> wrote:
The referenced sequence records numbers whose largest prime factor falls short of the square root of the number.
The graph of the sequence makes it clear that these numbers have a well-defined asymptotic density.
Following a citation to HAKMEM 29 (Schroeppel), we find a claim equivalent to saying that this density is 1 - ln 2 ~ 0.30685
But empirically, the density looks considerably smaller, more like 0.27.
The problem can't be whether or not to include exact squares of primes, because those have asymptotic density 0.
We have one more reported reference, to Exercise 28 on p. 26 of Tenenbaum and Wu, Théorie analytique et probabiliste des nombres, solution on p.
But I don't have a copy, and my French isn't up to the task of understanding their solution. Can somebody check it and compare it with Schroeppel's result (which, typically for HAKMEM, is stated without proof)? Rich, do you remember your approach? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun