User:Alonso del Arte/Is this sequence interesting

For a long time I vacillated between putting this on my "user space" or on the "main space." In the end I opted for my user space because these are my opinions. If I consider a newly submitted sequence interesting, that does not guarantee that it will be reviewed and approved quickly; nor if I find it uninteresting, it does not guarantee it will be rejected. However, anyone with the slightest doubt their sequence might be considered uninteresting would do well to read these opinions and anticipate objections I and others might have.

The question Is this sequence interesting? often is a subconscious shorthand for two other questions:

1. Do other people find this sequence interesting? and
2. Should I send it in to the OEIS?

No page written beforehand could possibly answer these questions satisfactorily for any sequence that might come up. But, if the sequence is found interesting, can be formatted for the OEIS and it is not already in the OEIS, it should be sent in.

Also, there is the awareness that any sequence that is truly of interest is already in the OEIS in some way or another, if not with an entry of its own, then as part of another sequence (such as a diagonal in a number triangle) or a merging of two or more sequences. If this is the case with a sequence you're thinking of sending in, then it becomes a question of whether or not the sequence is interesting enough to merit its own entry.

Interesting and ready for the OEIS

• A simple function or procedure leads to unexpected results.
• A neat illustration of a well-known theorem.
• A new result from the literature. I didn't think of this one at first (see User:Charles R Greathouse IV/Is this sequence interesting).

Interesting but not ready for the OEIS

• A sequence with way too many holes even though lots of people have studied it. If you can't rub more than two consecutive terms together, can't say if it's finite or not, can't say whether there might be cases needing the provisional clause "or x if there is no solution," don't know what value to use for the no solution cases (0? –1? something else?), then it's probably better to hold off on sending it in.

Not interesting, probably not for the OEIS

• Primes of the form $10^{k}+c$, where $k$ is an iterator and $c$ is some odd integer. So, we're dealing with primes that, once $10^{k}>10c$, consist of a 1 followed by a seemingly random amount of 0s and then the digits of $c$ (or, if $c$ is negative, a bunch of 9s followed by the digits of $10^{1+\log _{10}c}-c$.
  • Exception that proves the rule: Primes of the form $10^{k}+1$. Any other $c$ seems extremely arbitrary.
• Ditto $k$ such that $10^{k}+c$ is prime.
• $f(10^{n})$, whether or not the function has any deep connection to the number 10. Such a sequence seems almost an accidental consequence of base 10 having become our base of choice. I know you're thinking of a counterexample right now, and I've thought about it, too. I'll get to it in a little bit.
• A very complicated function or procedure leads to somewhat predictable results. For example, take the even digits of a number and reconstitute them as a new integer, square that integer, take the odd digits, blah, blah, blah (twenty steps omitted), take every third integer and see if that results in a semiprime. At what point did I lose you?
• Linear recurrences that don't answer any question. Just because the Fibonacci numbers and the Lucas numbers are so interesting doesn't guarantee that any recurrence relation $a(n)=a(n-1)+a(n-2)$ with arbitrary initial values will also be of interest.
• Small modifications of core sequences to solve non-problems. For example, one day "John Smith" was very bothered by the fact that the number 2 occurs at an odd position (1) in A000040. But if there is one thing John is even more adamant about is that the number 1 is not prime: anyone who even suggests it is a complete moron. So John came up with a solution: 9901, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, ... For some reason, though, John is not bothered by the fact that now 5 occurs at an even position (in fact, apparently all primes in A031368—but wait a minute: how was 9901 chosen to head the list? And is 9901 supposed to occur again as we exhaust the 4-digit primes?)
• Arbitrary transformations of sequences already in the OEIS. For example, John proposes A555001, the Schmuckelberg transform of A235001.
• Characteristic functions of other sequences. Essentially, a bunch of 0s with a few 1s at positions corresponding to numbers in other sequences.
  • Exception that proves the rule: A010051, characteristic function of primes: 1 if n is prime else 0.
• Number triangles that merely shuffle other triangles. Especially if the 'source' triangle was only mildly interesting to begin with. But it's also possible to come up with boring triangles based off Pascal's triangle.
  • Exception that proves the rule: A034851, Lozanić's triangle.
• Primes of the form $f(g(h(k)))$. Maybe we care about primes of the form $f(k)$ or $g(k)$ or $h(k)$. We might even care about primes of the form $f(g(k))$ or $f(h(k))$ or $g(h(k))$ or $h(f(k))$, etc. But $f(g(h(k)))$ just seems so contrived.
• The output of a pseudorandom number generator. Such as you might get from a QBasic program run.
  • Exception that proves the rule: A061364, pseudo-random numbers: a (very weak) pseudo-random number generator from the second edition of the C book. (This falls under one of the categories in the next section).

What a lot of these boil down to is aimless number crunching: putting numbers through a bunch of different processes without really caring what comes out of it, without wanting to prove a theorem nor discover some unexpected identity. If that's the kind of sequences we really wanted, we could just use a random (ahem, pseudorandom) number generator to come up with these and eliminate the need for human contributors.

Not interesting but maybe should be in the OEIS

Any respectable lexicographical work must of necessity contain certain entries that are not terribly exciting but yet must be included. Does any author of an English language dictionary start such a work because he's excited to define the word "the"? Yet "the" must be in such a dictionary.

• Very predictable but extremely useful sequences. Like A000027 (the "the" of the OEIS) and A000035. But probably all such sequences are already in the OEIS by now.
• Sequences that appear in books, and...
  • ...are erroneous. Therefore the OEIS must have an entry for them so as to be able to point people in the right direction. There is usually nothing interesting about a typo (the author forgot to carry the 1 in a computation, the typesetter's finger slipped on the keyboard, whatever), but the duty of the lexicographer outweighs such a lack of interest.
  • ...are correct. For example, a book on the Riemann zeta function probably has a table showing $\pi (10^{n})$ compared to some functions that estimate it. I still don't think this is a terribly interesting sequence, but, being a useful illustration, appears in many books, and therefore it should be in the OEIS (and in fact it is, see A006880).
• Sequences demanded by a very clear-cut analogy. Suppose that the OEIS has composite Lucas numbers (A172159) but not composite Fibonacci numbers. If you notice such a deficiency, you should send it that sequence even if you know nothing interesting about it. (This is just for the sake of example: A090206 has been in the OEIS for quite some time, and the sequence is interesting, at least in my opinion). Commonsense should dictate how far to take the analogy. If the OEIS has base 20 Horace numbers, that doesn't necessarily mean it should also have base 21 and base 22 Horace numbers as well.

Of non-general interest, not for the OEIS

"General interest" has always been given as a criterion for inclusion in the OEIS. Maybe "non-general interest" goes without saying. But what the heck, I will say it:

• Of only private interest. e.g., your grandmother's height in inches at her nth birthday. Probably reaches a record before a(20), stays the same until about a(80), at which we see a gradual decline due to osteoporosis.
• Of only local interest. A rare exception: A000053.
• Of only short-time interest. e.g., a biography of Leonhard Euler mentions Edward Waring on pages xii, xiv, 27, 331, 348, 355 and 407. Aside from the problem of distinguishing pages in the front matter from pages in the main body of the book (we can't enter Roman numerals in the data field of an OEIS sequence entry), of what interest is this to someone not researching Waring? Or to someone researching Waring but using a different edition of the Euler bio?
• And other categories I haven't even thought of.

The next step

• You are convinced the sequence is interesting. Look it up in the OEIS as if you were convinced it is already in. If you don't find it, then send it in (it might be a good idea to go through some kind of checklist, like this one). If you do find it, see if there is some comment or formula you can add to the entry.
• You are convinced the sequence is not interesting. I would at least write it in my notebook. You never know, a few years down the line you might come back to the same general problem.
• You are still unsure. Then it's time to go on a forum like SeqFan and ask "Is this sequence interesting?"

And after that

Sometimes you will come up with a sequence that everyone finds interesting and you will get lots of suggestions for related sequences, and variations. On the one hand, you want to follow up on each those and consider each of them in turn. But on the other hand, you don't want to go to the extreme of trying to think up every possible little variation on that one sequence people liked. "There he goes again with yet another spin on Horace numbers!"

For such related sequences, if they don't appear all that interesting in their own right, the question should be: is this an obvious variation or related sequence that anyone pondering the original sequence might come upon?

See also

• User:Charles R Greathouse IV/Imp