Or The Novice’s Guide To Achieving Mathematical Immortality

This is a guest post from Barney Maunder-Taylor.

A great way to achieve mathematical immortality is to solve an outstanding open question, like determining if $\pi +e$ is rational or irrational, or finding a counterexample to the Goldbach Conjecture. But for most of us, a more realistic approach would be to contribute a new sequence to the Online Encyclopaedia of Integer Sequences, the OEIS. This is the tale of one mathematician’s quest to do just that – with ideas to help YOU to contribute a sequence of your own.

Based on a collection started in 1964 by N. J. A. Sloane (still the Chairman today), the OEIS now contains over 378,000 sequences of whole numbers, with frequent new submissions. It attracts thousands of users every day: enter a few terms of your favourite sequence, hit search, and BLAM! Up pop 20 or 30 terms plus all contexts in which it’s known to appear. You may even get a formula and/or Python code – it’s like a Christmas stocking of number knowledge.

All the sequences you’d expect are there (primes, squares, cubes, triangular numbers, Fibonacci numbers, etc.,) plus a vast collection of wacky and esoteric ones (digits in the decimal expansion of $\sqrt{51}$; the number of letters it takes to spell the integers in French; number of blot hitting numbers when the blot is $n$ points away in a game of Backgammon, and hundreds of thousands more).

The challenge, then, is to find a sequence that’s NOT already listed in OEIS. The rules: it has to be mathematically meaningful (although you’re likely to still get a hit if you put in a few random numbers and hit search), and we need to know at least 4 terms.

How hard can it be to contribute to this behemoth of mathematical knowledge? Might we try the square triangular numbers? Tap tap tap – obviously there. Too easy. How about numbers such that $n$ and its reversal are both multiples of 17?  [272, 323, 595, 646 search] Yup, there. Numbers that are simultaneously triangular and square pyramidal? [1, 55, 91 search] – the sequence ends after 5 terms – and of course it’s there.

Four years of blood, sweat and failure later…

It’s the 2024 MathsJam Gathering. Present are some of the finest, most generous and good-humoured maths brains you’ll find… plus me: a fun-loving Primary Maths Communicator. Katie Steckles circulates a puzzle: can you find two successive integers both of whose digit sums are a multiple of 7? AND THE ANSWER: …don’t know, let’s wait until the next East Dorset MathsJam. The following week, regular Tom B quickly finds 69,999 and 70,000 (digit sums $6+9+9+9+9=42=6\times 7$, and $7+0+0+0+0=7=7\times 1$, so both multiples of 7 as required). We’re pretty sure that’s the smallest such pair.

Hang on, back up please…

Why 7? What if we generalise and change 7 to $n$? OK: $n=1$ easy. $n=2$: yep, also easy. $n=3$, hmm, not sure that’s going to work, a few more terms, an OEIS search… it’s not there!! And so was born sequence A378119: defined by $a(n)=$ the smallest positive integer $k$ such that the digit sums of $k$ and $k+1$ are both divisible by $n$ (or $-1$ if no such pair exists; apparently this is the standard OEIS marker for non-existent terms). Pause now if you’d like to reread that definition and then figure out the first few terms. You should expect bad things to happen if $n$ is a multiple of 3 (or else, due to the digit sum test for divisibility by 3,  we would have two successive integers which are both multiples of 3 – which is clearly impossible). All of the other terms seem to exist. Of course, you can type A378119 into the OEIS search bar — or click this link — for the answers.

Submitting a new sequence to OEIS, thereby both contributing to the wealth of human mathematical knowledge AND looking really cool at your next MathsJam in the pub – is surprisingly straightforward:

1. Register for an OEIS account.
2. Login and select contribute, or visit the new sequence page. The instructions are very clear. For example: “enter the initial terms of the sequence here (required). Entries usually give at most 260 characters, but check the “style sheet” for extra guidance.”
3. Submit! An email confirms that your sequence has been ALLOCATED – with its own A number.
4. Over the coming days or weeks your sequence will be looked over by moderators. Even so much as a space out of place will put you in the scrutiny firing line, so take care!
5. Your sequence goes through the stages of APPROVED and EDITING, usually with some ping-ponging back and forth, until…
6. (email arrives): Alice Person PUBLISHED your changes to sequence A378119.

I anticipate that you will now all be scurrying to publish your first OEIS sequence. Here are two that appear to have not yet been claimed. Go ahead, they’re all yours – enjoy!

• Cube numbers that contain at least one 7.
• $a(n)=$ the smallest integer $k$ such that the digit sums of $k$ and $k^2$ are both multiples of $n$.