[seqfan] Re: Benford's law and the OEIS

[Neil Sloane]

PS
Forgot to mention this:
A Berger, TP Hill, E Rogers - URL: http://www.benfordonline.net, 2009
Benford online bibliography - very useful site

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 Thu, Feb 9, 2017 at 8:10 PM, Neil Sloane <njasloane at gmail.com> wrote:

> Dear Sequence Fans,
>
> A sequence a(n) satisfies Benford's law if for any fixed d in the range 1
> to 9, the limit
>
> {no. of k (1<=k<=n) such that a(k) begins with d} / n
>
> exists and equals log_10(1+1/d).
>
> One must be careful, for A000027, a(n)=n, the limits do not exist,
> so Benford's law is not satisfied.
>
> I have created an entry in the OEIS Index for Benford's law, which
> lists both sequences that satisfy it and those that don't,
> and those where the answer is not known (A003095 is a famous open
> question),
> and other related sequences.  (Only a selection of all these, of course,
> or else there would be too many entries.)
>
> The Index entry is
> https://oeis.org/wiki/Index_to_OEIS:_Section_Be#Benford
>
> The status of many of the core sequences is known (to me):
> Benford: 2^n, n!, Fib_n, Lucas_n, partitions(n), Pascal's triangle A007318
> Not Benford: n, n^2, n^3,  etc, prime(n), and I think binomial(n,k) for
> fixed k
>
> There is a paper by Hurlimann,
>
> Hürlimann, W (2009). Generalizing Benford’s law using power laws:
> application to integer sequences. International Journal of Mathematics and
> Mathematical Sciences, Article ID 970284. DOI:10.1155/2009/970284.
>
> where it is clear (although he does not mention the OEIS) that he took a
> selection of sequences from the OEIS, and ran numerical tests to see which
> were Benford. There are two in particular that he mentions, the Catalan
> numbers (A000108, binomial(2n,n)/(n+1) ), and the Bell numbers, (A000110),
> which appear to be Benford
> according to his data, and which must surely be Benford, although I
> haven't yet seen a proof for either. For now I've given them the benefit of
> the doubt.
>
> But there are others that H. mentions where the status is less clear.
> There are a huge number of articles and web pages and books on the subject,
> and I have only looked at a few of them.
>
> There is a 2015 book by Berger and Hill that I've ordered that I hope will
> clear up some of these questions.  Berger and Hill have a nice article in
> the Feb issue of the Notices
> of the Amer Math Soc, which is what got me interested in the question. I
> can send a copy
> if you don't have access to it.
>
> If anyone wants to help, by updating the index entry (and at the same time
> adding appropriate
> comments to the entry for the sequence - see A282022 for an example)
> please do so.
> But remember, it is not enough to check the ratios after 10000 terms say.
> You also have to show that the limits exist.
>
>
>
>