What makes a sequence beautiful?

[Jonathan Post]

Many great mathematicians have written about the beauty of
mathematics. I need not bibliographize the many papers and books.
Many recent essays on the subject have been published by people as
deep and brilliant as Terry Tao and John Baez.

For Seqfans, and users of OEIS, the issue of mathematical beauty has a
specific flavor: What makes a sequence beautiful?

There are many answers. Here are a few that I've pondered.

(1) A sequence that tells us something about a beautiful number can be
beautiful. Hence all the sequences about pi, e, gamma, and the like.

(2) A sequence that enumerates beautiful objects can be beautiful.
Hence all the sequences that enumerate groups, semigroups,
permutations, partitions.

(3) A sequence that unexpectedly connects two objects that were not
expected to be connected can be beautiful. The beauty is related to
both the surprise and the retroactive inevitability.

(4) Numbers do not exist in isolation.  A sequence shatters the
isolation by relating a set of numbers. That relationship, whether an
enumeration, or indexing, or recusrsion can make the sequence
beautiful.

(5) Sequences do not exist in isolation.  Hence there are so many
sequences which are explicitly transformations of other sequences.
This can be beautiful.  A sequence can introduce a new transformation.

(6) A sequence can be so important that it is beautiful for its
centrality in sets of sequences.  That is part of what is meant by
"core."

I can gave many more such classifications of sequence beauty.  njas is
teaching us something deep when he identifies a sequence as "nice."
That's a carrot.  The sticks are "less" and "probation."  Yet it is
the same lesson, from different directions.

Primes are beautiful. Yet it is a lesson from njas that "primes of the
form xyz" may be a very much less interesting and ugly thing.

I have annoyed some people by how many sequences that I have submitted
about semiprimes.  Worse, from that standpoint, that I have dragged
not only associate editors but also others to do the same.  Am I a one
trick pony for submitting so many that say "this seq is to semiprimes
A001358 as Axxxxxx is to primes A000040"?

For those who dislike these, it seems even more artificial to submit
sequences about k-almost primes which are analogues of sequences about
primes.  We know that njas dislikes the very term "k-almost prime."

Let me comment that, in a sense, everything about integers is
determined by the primes.  This comes from the Fundamental Theorem of
Arithmetic, unique prime factorization. Hence semiprimes tell us
nothing new that isn't already implicit in the primes.

However, there is another point of view which I have found seductive.
Okay, we know that every integer beyond 1 has a unique prime
factorization.  We know how totally multiplicative funtions are
determined by their evaluations over primes.  But the prime
factorization adds a structure to the integers, and a beautful
structure.

To me and some of my coauthors, a beautiful structure is the array
whose k-th row is the k-almost primes, i.e. the integers with exactly
k prime factors (with multiplicity).  The 1st row is the primes.  The
second row is the semiprimes. And so on.  It is also obvious what the
columns are.

To me, in the context of that structure, the relative "core" sequence
is the main diagonal, A101695, a(n) = the n-th n-almost prime. That
and A078841 are relatively core, being features of the entire array,
and not merely a row or column.  hence to me, those are beautful
sequences.

Hence, to me, some apparently "less" interesting and articial sequence
ar straighforward explorations of the relative core sequences, and
hence, if not beautiful, at least not "less" beautiful, i.e. at least
not ugly.

Now, there are are least 5 kinds of "truth" that each have their own
notions of "proof", of deduction, of evidence, of social protocol:

(1) Axiomatic Truth, the beating heart of pure Mathematics, from Euclid on.

(2) Empirical Truth, from the Scientific methods, and, more recently,
from Experimental Mathematics a la Borwein et al.

(3) Legal-political Truth.

(4) Aesthetic Truth.

(5) Revealed or religious or mystical Truth.

No two of these are the same, and much agony comes from the
philosophical category error of confusing one with another.
Legislating the value of pi. Outlawing an art form. China enforcing
laws about Tibetan reincarnation. Seeking beauty in a test tube, or
equations in prayer (unless you're Ramaujan).

There are sequences that come from (2) Empirical truth, about the
elements of the preiodic tables, or the orbits of planets, or the
like.

There are sequences that come from legal-political truth, as the
sequence of the number of nations in the United Nations.

It is, to me a breakthrough, that we can listen to sequences, whether
originally from great melodies of Musics, or discovering musical
beauty in sequences.

It is sometimes a surprise to me how beautiful the graph or scatter
plot of a sequence can be, fing a 2-D painting in a sequence not
obviously beautiful in 1 dimension.

A song is beautiful or ugly to you regardless of what the composer,
singer, or critic says.  Same for a painting, a sculpture, a building,
or a sequence.

Except that one grows and changes over time. What first seems discord
can become beautiful.  People stormed out of Beethoven symphony
premiers, or stormed out of art museums, outraged, and we now wonder
why.

Hence there is little value in the submitter of sequence complaining
that the sequence is not "less" but really beautiful in the eye of its
creator.

We can't individually be sure if our tastes are evolving or devolving.
 Parents sometimes stop, mute in midsentence, when they tell their
teenager: "That's not music you're listening to, that's just noise;
why, in my day we had great music..." when they suddenly recall their
own parents saying that to them a generation earlier.

Where is mathematics going?  No individual can know, or say.  The
History of Mathematics will make that all so clear, retroactively.  We
must live mathematics forwards, but understand it backwards.

What makes a sequence beautiful?

I do not have answers.  But I appreciate the conversation that we are
all having through OEIS and seqfans.  It is more than a conversation.
It is a symphony.