[seqfan] Benford's law and the OEIS

[Neil Sloane]

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.