The 82000 sequence is still generating interest.

Here’s a great Numberphile video about the sequence from mathematician James Grime:

[youtube http://www.youtube.com/watch?v=LNS1fabDkeA]

The sequence has also made it into the Online Encyclopaedia of Integer Sequences, as A258107!

I have also found an early mention of the special properties of 82000, in this exchange on a French puzzle forum, from October 2008. In the thread, user “dhrm77” challenges others to identify “a number greater than 1, which is written only with 0 and 1 in bases 2, 3, 4 and 5.” More excitingly, he claims that “this number is unique”, which made me hopeful that there might be a proof of this fact further down the thread. No such luck though, as “dhrm77” later admits (in post #11) that he has no proof of uniqueness, and discusses the way in which the chance of finding a solution becomes increasingly small as the size of the numbers you are checking increases.

The final comment (#13) refers users to OEIS sequence A146025, which “dhrm77” (presumably the “Daniel Mondot” cited as the sequence author) has apparently submitted and had accepted in the OEIS. Whether “dhrm77” discovered the interesting property of 82000 himself, or whether he got it from some earlier source is not clear.

Daniel MondotTo answer the last sentence, yes I found this number or sequence myself.

A few years ago, I started this quest of finding series of numbers that used a limited number of digits when written in an increasing number of bases.

I started to check the numbers that only uses digits 0 to 9.

If we limit our search to base 17 or less, we quickly find out that there are an infinite number of numbers that matches the search.

If we then add base 18 to the limitation, not count digits 0 to 9 themselves, there are only 10 numbers that match the criteria (http://oeis.org/A131646)

If we then add base 19 to the limitation, not count digits 0 to 9 themselves, the list goes down to 2: 19 and 20.

If we then add base 20 to the limitation, not count digits 0 to 9 themselves, the list goes down to just 1:20 itself.

Adding one more base after that, and there are no more numbers that match.

So I got very interested in this narrow band between the “infinite number of numbers” and “no number at all”, (except for the single digits themselves), where the list of numbers appears to be finite.

I wrote a multitude of programs to hunt these numbers. The first version was just brute force, slow a didn’t go very far. Then I improved it to the point where I could hunt them all the way to 2^65520, in the fraction of the time it originally took me to scan up to 2^64

I wrote the latest versions to hunt series of numbers up to base 64.

I didn’t push the search to 2^65520 for every sequence. The highest sequence where I went that far is sequence #A146027 in OEIS and it took 309 hours.

For sequence #A131646, I only went to 2^20356 and it took 521 hours, 43 minutes to get there.

Unfortunately, I haven’t worked on these since December 2008, and didn’t make any more progress.

If you are fascinated by these sequences, and would like more information, please email me (gmail).

thomasPost authorThanks for the detailed comments. Great to hear from you on this. It is a fascinating area and it would be great (though probably intensely difficult) to push for some real theoretical results. Not sure I have the time or the inclination to do the pushing though!

Daniel MondotI have a theory, quite simple as a matter of fact, concerning the narrow band. I am able to predict to some degree one of the limit of the band.

i.e. for a particular set of digits (e.g 0 to 9) and a particular set of bases (e.g. 2 to 18), I can predict if the number of numbers that matches the criteria (the “sequence”), is finite or infinite.

And I have a table showing for each group of bases 2 to 46 and digits 0 to 36 what the actual results are.

But I no longer have a website to post it on.

Some day I will work on it again….

I believe it is somewhat pointless to do a search for over a million digits, as someone on Reddit claims to have done because, for some sequences, as you increase the number of digits in your search, the odds of finding new large numbers that match the rule multiplied by the number of number existing in the next group are decreasing so much, it become extremely unlikely to find new ones.

For some other sequences however, like the sequence for numbers that only use digits 0 to 51 in bases 2 to 75, the largest number found has 3421 digits in base 10, and it could be interesting to push the search a little further than for other sequences that become “dry” much quicker.

thomasPost authorInteresting. Is your argument anything like the one that I make towards the bottom of this post:

http://www.mathistopheles.co.uk/maths/covering-all-the-bases/solution-covering-all-the-bases/

Daniel MondotYes, I believe so.

You made the same observations, and did the math slightly differently than I did.

And I considered the general case, not just the 0 and 1 case.

But essentially, it’s the same argument.