# What is an example of a sequence which “thins out” and is finite?

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?

An example would be the narcissistic numbers, which are the natural numbers whose decimal expansion can be written with $$nn$$ digits and which are equal to sum of the $$nn$$th powers of their digits. For instance, $$153153$$ is a narcissistic number, since it has $$33$$ digits and$$153=13+53+33.153=1^3+5^3+3^3.$$Of course, any natural number smaller than $$1010$$ is a narcissistic number, but there are only $$7979$$ more of them, the largest of which is$$115132219018763992565095597973971522401.115\,132\,219\,018\,763\,992\,565\,095\,597\,973\,971\,522\,401.$$