TITLE: What is an example of a sequence which "thins out" and is finite?
QUESTION [80 upvotes]: When I talk about my research with non-mathematicians who are, however, interested in what I do, I always start by asking them basic questions about the primes.  Usually, they start getting reeled in if I ask them if there's infinitely many or not, and often the conversation remains by this question. Almost everyone guesses there are infinitely many, although they "thin out", and seem to say it's "obvious": "you keep finding them never mind how far along you go" or "there are infinitely many numbers so there must also always be primes".  
When I say that's not really an argument there then they may surrender this, but I can see they're not super convinced either.  What I would like is to present them with another sequence which also "thins out" but which is finite.  Crucially, this sequence must also be intuitive enough that non-mathematicians (as in, people not familiar with our terminology) can grasp the concept in a casual conversation.
Is there such a sequence?

REPLY [4 votes]: Divisible prefixes
Here's one  you might find useful:

Numbers like 1200549600848 have the property that the first $k$ digits form a number divisible by $k$. For example, 12 is divisible by 2, 120 is divisible by 3, 1200 is divisible by 4, 12005 is divisible by 5, and so on.
If you're already talking with someone about primes, this divisibility property might build on what they already understand and therefore be relatively easy to grasp.
It turns out there are only finitely many numbers with this property. 
In particular, you can prove that 3608528850368400786036725 is the only 25-digit number with this property. And this number can't be extended to a 26-digit number with this property. It follows that there are no higher numbers with this property (because if there were, the 26-length prefix of such a number would have this property).