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

[Allan Wechsler]

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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>
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