[seqfan] Re: Typo in B-file for A063539
jean-paul.allouche at imj-prg.fr
Fri Apr 3 18:29:01 CEST 2020
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
# GÉRALD TENENBAUM <https://www.belin-education.com/gerald-tenenbaum>, JIE
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:
by clicking on the cover page.
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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