A Strange Game With Digits
Exploring the Integer Sequence A359012
There is a certain kind of integer that seems determined to get lost and then rediscover itself.
Take a positive integer k and split it into two pieces x and y, so that k = concat(x, y). Now throw away the order of all those digits and permute them freely to make some new number, which we call P(x, y). Finally, glue everything back together as x P(x, y) y and read the result as a decimal string.
Most of the time, k disappears in the chaos of the permutation. But every so often, something odd happens: the original integer k reappears as a contiguous substring somewhere inside x P(x, y) y. When that happens, we keep k. These self-reappearing integers form the sequence A359012, catalogued in the On-Line Encyclopedia of Integer Sequences (OEIS).
What makes A359012 interesting right now is not a closed-form formula or a deep theorem, but its shape. I have been building an interactive Sequence Atlas that reads a live dataset of all currently known terms, and turns it into a landscape: dashboards, histograms, gap plots, and a row-by-row browser where you can inspect individual cases. You can think of it as a tiny observatory for a very specific kind of numerical weirdness.
The Field Guide: What the Atlas Records
For every integer k in the sequence, the atlas records a few key pieces of information. First, there is the successful split (x, y): the specific way of cutting k’s decimal digits into a left part and a right part that made the trick work. Then there is the witness P(x, y): a permutation of those digits for which k genuinely reappears as a substring inside x P(x, y) y. Beyond that, the atlas records the position inside the witness where k was found, a normalized depth value between 0 and 1 (where 0 means the very start and 1 means the very end), and an expansion ratio defined as the length of the witness divided by the length of k, telling you how much the witness inflates compared to the original number.
Take k = 318 as a small example. One valid split is x = 31 and y = 8. If you find a permutation P(31, 8) of the digits {3, 1, 8} such that the string “318” appears inside “31” + P + “8”, then 318 earns its place in A359012. The atlas records which permutation does the job, how far into the combined string “318” sits, and how long the full witness is relative to three digits. Even for small numbers like this, the answer is not obvious at a glance, and that is precisely what makes the database worth having.
Curiosities in the Current Dataset
Once you start looking at the atlas as a whole, certain facts jump out immediately.
The first one is about zeros. Out of the 712 unique terms currently in the dataset, 708 have witnesses that end in a trailing zero. In other words, when you scramble the digits of x and y to form P(x, y), the permutation value almost always collapses into something that ends in 0. This trailing-zero dominance is so strong that non-trailing-zero witnesses feel like genuine anomalies rather than statistical noise.
The dataset flags a handful of those anomalies explicitly. The term k = 692 comes from the split x = 69, y = 2, and its witness ends in 2 rather than 0. Another outlier is k = 55383 with split x = 5538, y = 3, whose witness ends in 6. Two more terms, 218244 and 723293, also manage to escape the trailing-zero pattern. With only four such rebels against hundreds of trailing-zero witnesses, they stand out as genuine exceptions in the landscape.
Digit repetition is another recurring theme. Of the 712 terms, 572 contain at least one repeated digit. All-distinct-digit terms are the exceptions rather than the rule. That makes some intuitive sense: repeated digits give more combinatorial flexibility when you shuffle everything into a permutation, and that extra flexibility may help the original pattern sneak back in as a substring.
There are also a few charming special cases that the atlas highlights: palindromes. At the moment, only four terms in the search window happen to be palindromes: 9999, 29092, 343343, and 805508. These are fun to inspect individually in the browser because you can see how a symmetric decimal pattern manages to find itself again inside a scrambled witness. Even within this small family, the splits and witness lengths vary, so there is no single obvious recipe for building palindromic terms.
Edge behavior is surprisingly rare as well. The atlas tracks whether a term appears at the very start (prefix) or very end (suffix) of its witness, and the numbers are tiny: just 3 prefix hits and 1 suffix hit in the entire current dataset. Almost every term lands strictly in the interior of its witness. If you think of the witness as a long line of digits, terms prefer to live somewhere in the middle rather than at either edge.
A different kind of curiosity lives in the gaps between consecutive terms. The largest inter-term gap currently observed is 26,763 integers. On a gap chart, this shows up as a tall spike: a long desert where no integer qualifies. It is a visual reminder that A359012 is extremely sparse. Even within the range already explored (up to 997,398), huge stretches of the number line sit completely empty.
Finally, there is a structural absence that may or may not persist as the search continues: no term has yet been observed with two different splits (x, y) that both produce valid witnesses. For every k in the dataset, exactly one split works. This could simply be a coincidence in the current search window, or it might point to a deeper constraint that makes multiple splits rare or impossible. Either way, it is the kind of negative fact that is very easy to miss without a clean, queryable dataset.
The Large-Scale Landscape
Zooming out from individual terms to the whole dataset reveals a different kind of structure: a shape, rather than a list.
Term lengths in the current dataset cluster heavily around six digits. Three-, four-, and five-digit terms are much rarer in the current search window, which suggests that whatever combinatorial conditions make the substring trick possible become easier to satisfy at certain digit counts. Whether that clustering persists as the search goes deeper into larger numbers remains an open question.
The geometry of splits shows similar structure. A heatmap of the pair (length of x, length of y) reveals that certain split shapes are strongly preferred. Not every way of cutting k in half is equally productive: some geometries yield witnesses reliably, while others appear almost never. This is not predicted by a simple counting argument, which makes the heatmap one of the more visually striking panels in the atlas.
The growth rate of the sequence is sublinear. On a log-log plot, the cumulative count of terms up to N grows roughly like N raised to the power 0.816. This confirms that A359012 is genuinely thinner than a linear sequence: the fraction of integers that qualify shrinks as you go further. The sequence is not just sparse, it gets sparser.
The depth distribution, on the other hand, is surprisingly flat. Across all known terms, the position where k appears inside its witness (normalized from 0 to 1) shows no overwhelming preference for start, middle, or end. Terms land roughly evenly across the witness, with only small bumps hinting at any deeper bias. This near-flatness is itself an interesting fact: it means the permutation process does not systematically push k toward one side of the constructed string.
Try It Yourself
The Sequence Atlas is built to be browsed, not just read. A few entry points are worth exploring directly.
The dashboards give a bird’s-eye view: term-length histograms, split-geometry heatmaps, gap charts, depth distributions, and the log-log growth curve. These are the best place to start if you want the overall picture without committing to individual rows.
The row browser lets you filter and sort the full table. You can search for your favorite three-to-six digit number and see whether it appears; filter to palindromes or primes and inspect their witnesses; or sort by expansion ratio to find the most inflated cases. The outliers are often the most interesting starting points.
A few concrete challenges worth trying: look for a term whose depth is very close to 0 or very close to 1, and work out by hand why it lands at that edge. Try to construct a six-digit term manually by choosing a split and enumerating permutations. Check whether any prime in the dataset has a witness that is also prime. None of these are guaranteed to lead anywhere, but they are the kind of hands-on questions that turn a passive read into an active exploration.
Open Questions
A359012 sits at an unusual point in the landscape of integer sequences: it is defined by a concrete and checkable rule, yet no one has yet explained from first principles why it has the shape it does.
Several questions feel tractable but remain open. Can you prove that the density of terms decays, or pin down the growth exponent more precisely than the empirical 0.816? Is there a structural reason why almost all witnesses end in zero, or is it a coincidence of the current search range? Are there infinitely many palindromic terms, or will the four currently known remain lonely? Can two different splits ever both work for the same integer k, or is there a theorem ruling that out?
These questions do not require heavy machinery to state, only a curious mind and perhaps a little code. The atlas is there to help: it turns the raw sequence into something you can look at, poke, and ask questions of. Whether you end up answering one of those questions or simply spending an hour chasing an interesting row in the browser, that seems like a worthwhile way to spend time with a strange corner of arithmetic.
