[seqfan] Some comments about the OEIS from Hal Switkay (forwarded with his permission)

[Neil Sloane]

Dear Seq Fans, Yesterday Hal Switkay sent me an email which I thought was
interesting enough to post here. I haven't included his spreadsheet - if
you want a copy, write to him at hswitkay at hotmail.com

(Start quote)
I think a lot about the many different ways in which natural numbers can
have many divisors: highly composite, deeply composite, superabundant,
their refinements, and several interesting non-trivial variations I have
made of these concepts. It seems that the most restrictive such subsets
(demanding the highest levels of achievement) usually include the
following: 1, 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440.
[This is now A328549] This is the intersection of colossally superabundant,
superior highly composite, and deeply composite; if I intersect with
sparsely totient as well, the list narrows to 2, 6, 12, 60, 120.

I began to wonder about the OEIS corpus as a statistical entity and as a
metric entity. I wondered about the frequency of n-grams (ordered sequences
of length n) in OEIS, and about a distance function on pairs of sequences.
I have a rather complex semi-metric measuring the distances between two
infinite sequences; I don't know if it satisfies the triangle inequality.

The attached spreadsheet [omitted here] measures the OEIS frequencies of
some short sequences with small sums. I created some measurements on these
sequences as I went along: weight = sum + length; then I found weight +
length to be more useful. The number of sequences of a given weight is a
power of 2; the number of sequences of a given (weight + length) is a
Fibonacci number. I am considering a third measure as well:
(a(1)+1)(2a(2)+1)(3a(3)+1)...

The frequencies were gathered over two consecutive days in which the OEIS
total sequence count changed slightly, so there isn't 100% exact
correspondence for all of these frequencies, but I think it is close
enough. I wanted to know the most frequent number - apparently 1; the most
frequent bigram - apparently 1,2 - and so on.

It occurred to me that the most frequent trigram is not necessarily going
to correspond with a concatenation of the most frequent bigram 1,2 with
another frequent bigram that connects with 1,2 at the beginning or the end.

I am very curious to see the most frequent 30 integers and their
frequencies; the most frequent 30 bigrams and their frequencies; the most
frequent 30 trigrams and their frequencies; and so on. What types of
sequences tend to be of more interest to OEIS users - numbers with many
divisors, numbers with few divisors, or sets of numbers with no consistent
divisibility properties?

(End quote)