[seqfan] Re: Typo in B-file for A063539

Sat Apr 11 22:37:03 CEST 2020

The convergence to an asymptotic factor of 3.26.. is actually very slow. So
far I have calculated the following values:
A063539(10^6)=3697669, A063539(10^7)=36519633, A063539(10^8)=360856296
It will take some time to get the next value A063539(10^9).

Hugo Pfoertner

On Sat, Apr 11, 2020 at 3:51 PM Neil Sloane <njasloane at gmail.com> wrote:

> Dear Jean-Paul, Allan, et al.
>
> I'm glad this mystery has been cleared up!  Could one of you add some
> comments (and references) to the appropriate sequence?
>
> 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 at gmail.com
>
>
>
> On Sat, Apr 11, 2020 at 4:53 AM jean-paul allouche <
> jean-paul.allouche at imj-prg.fr> wrote:
>
> > Dear Allan, dear all
> >
> > The result on the frequency is actually due to Dickman in
> > "On the frequency of numbers containing prime factors of a certain
> relative
> > magnitude", Ark. Mat. Astr. Fys. 22, 1930, 1-14.
> >
> > Most of the useful statements are actually available at
> > https://en.wikipedia.org/wiki/Dickman_function
> > (see in particular the sections "Properties" and "Applications").
> >
> > About the discrepancy between the values 3.26 and 3.76, this is quite
> > conceivable, in that the "true" value is asymptotic: it might well be
> that
> > the convergence is extremely slow so that 40000 is still "small" in the
> > asymptotic behavior. There is probably a formula with remainder that
> > could enforce this remark.
> >
> > best wishes
> > jean-paul
> >
> >
> >
> >
> > Le 10/04/2020 à 22:48, Allan Wechsler a écrit :
> > > Jean-Paul Allouche has a point.
> > >
> > > Empirically, the 3.76+ seems to be correct. In the B-file, a(10622) =
> > > 40000, and 40000/10622 = 3.76+.
> > >
> > > But Schroeppel claims in HAKMEM 29 that the probability that the
> largest
> > > prime factor of n exceeds sqrt(n) is ln 2 = 0.693147+. This would imply
> > > that the asymptotic value of a(n)/n would be 3.25889+, as stated by
> > > Allouche.
> > >
> > > Could the problem be due to the fact that https://oeis.org/A063539
> > insists
> > > on the largest prime factor being strictly less than the square root?
> > That
> > > is, could the discrepancy be attributable to numbers whose largest
> prime
> > > factor is exactly the square root? No, because these are just the
> squares
> > > of the primes, and their density is asymptotically 0. Just to be sure,
> I
> > > checked https://oeis.org/A048098, which includes the squares of the
> > primes.
> > > Here, the asymptotic value of a(n)/n is also close to 3.76 (3.7518, to
> be
> > > precise), and far from 3.26, the value predicted by Schroeppel.
> > >
> > > To resolve this puzzle, we should (a) hear from Rich Schroeppel about
> how
> > > the result was derived, and (b) inspect Tenenbaum and Wu, making sure
> > that
> > > they report the same result. Something is awry here. I can email
> > > Schroeppel, but I can't read French mathematics.
> > >
> > > On Fri, Apr 10, 2020 at 4:17 PM jean-paul allouche <
> > > jean-paul.allouche at imj-prg.fr> wrote:
> > >
> > >> Dear all
> > >>
> > >> I am not sure that my message below came through.
> > >>
> > >> Actually there is something more: the density being
> > >> (1 - ln 2), this implies that the n-th term of the sequence
> > >> is equivalent to Cn with C = 1/(1-ln (2)) which is about 3.259
> > >> (so that it is not 3.7642*n as indicated in the Formula Section.
> > >> Since I have a bad internet connection, it would be good if
> > >> someone could have a quick check and make the corresponding
> > >> changes in A063539.
> > >>
> > >> Many thanks in advance
> > >>
> > >> best wishes
> > >> jean-paul
> > >>
> > >>
> > >>
> > >> Le 03/04/2020 à 18:29, jean-paul allouche a écrit :
> > >>> Hi
> > >>>
> > >>> I asked Gérald Tenenbaum about the result stated by Schroeppel.
> > >>> He told me that this is, e.g., Exercise 28 (with proof) in his book
> > >>> with Jie Wu:
> > >>>
> > >>> # GÉRALD TENENBAUM
> > >>> <https://www.belin-education.com/gerald-tenenbaum>, JIE WU
> > >>> <https://www.belin-education.com/jie-wu>
> > >>> #
> > >>>
> > >>> #
> > >>>
> > >>>
> > >>>   Théorie analytique et probabiliste des nombres
> > >>>
> > >>>
> > >>>     307 exercices corrigés
> > >>>
> > >>>
> > >>> I double-checked: this is indeed Exercise 28 on Page 26, the solution
> > >>> can be found on Page 34. Everything is in French but this should not
> be
> > >>> a problem. Note that the first few pages of the book (including the
> two
> > >>> pages above) are freely accessible on the site of the publisher:
> > >>>
> > >>
> >
> https://www.belin-education.com/theorie-analytique-et-probabiliste-des-nombres
> > >>> by clicking on the cover page.
> > >>>
> > >>> best wishes
> > >>> jean-paul
> > >>>
> > >>>
> > >>>
> > >>>
> > >>> Le 02/04/2020 à 18:34, Allan Wechsler a écrit :
> > >>>> A063539 collects numbers whose largest prime factor is less than the
> > >>>> square
> > >>>> root. For example, 29925 = 3^2 * 5^2 * 7 * 19, and 19^2 is only 361,
> > >>>> much
> > >>>> smaller than 29925.
> > >>>>
> > >>>> An interesting feature of this sequence is that it has constant
> > >>>> asymptotic
> > >>>> density; HAKMEM item 29 (Schroeppel) identifies the density as (1 -
> > >>>> ln 2),
> > >>>> without proof.
> > >>>>
> > >>>> Because of this intriguing feature, it's interesting to look at the
> > >>>> graph
> > >>>> (which of course looks like a straight line), and this reveals an
> odd
> > >>>> blot
> > >>>> under the line, which I have traced to a typo in the B-file.
> > >>>>
> > >>>> A(7910) ought to be 29925 (the example I gave above), but is instead
> > >>>> given
> > >>>> as 9925, which should not be in the sequence because its largest
> prime
> > >>>> factor is 397.
> > >>>>
> > >>>> I wonder how typos like this can creep in -- the text of the B-file
> > >>>> ought
> > >>>> to be copied directly from program output, and never pass through
> > human
> > >>>> editorial hands which might drop a digit, as seems to have happened
> > >>>> here.
> > >>>>
> > >>>> (Also, the comments should include the slope of the line, ideally
> > with a
> > >>>> citation to someplace that proves the identity.)
> > >>>>
> > >>>> --
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> > >>>
> > >>> --
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