I know of various properties of numbers that are known to have no largest element (like natural numbers, primes, etc.) and I know of unproven conjectures that certain properties have a largest element (twin primes, etc.) but I don't remember examples of properties that have been proved to have a largest element, at least within number theory. I suspect there are some, so I thought I'd ask here.

Outside of number theory, I know of some geometric ones -- like the number of sides in a Platonic solid.

What's Special About This Number? may be a useful source.

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    $\begingroup$ oeis.org/… may give a start (though a lot of these are finite for stupid reasons "divisors of 24") $\endgroup$ – Hagen von Eitzen Sep 10 '18 at 6:23
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    $\begingroup$ Ah, OEIS with keyword:full (defined as: "full: The full sequence is given, either in the DATA section or in the b-file (implies the sequence is finite and has keyword "fini")". Good idea! $\endgroup$ – JesseW Sep 10 '18 at 6:25
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    $\begingroup$ fini might be even better and could include cases where finiteness is known, but the largest element is not. Still, there are nice finds, such as $\tau(n)=\sigma(n)\implies n\le 30$ and of course Heegner numbres $\endgroup$ – Hagen von Eitzen Sep 10 '18 at 6:30
  • $\begingroup$ Please add those as answers so I can vote for them! $\endgroup$ – JesseW Sep 10 '18 at 6:36

Here are two examples:

Factorions are numbers that are the sum of the factorial of their digits. For instance, $145 = 1!+4!+5!= 1+24+120=145$. The largest factorion is 40585.

Numbers that are the cube of their digit sum, such as $512 = (5+1+2)^{3} = 8^{3}$. The largest is 19683.

  • $\begingroup$ Very nice! The (purely stylistic) flaw I see in these is that they are dependent on the base used (I think). I'd love to hear of ones that aren't... $\endgroup$ – JesseW Sep 21 '18 at 5:49
  • $\begingroup$ Their entries are: oeis.org/A014080 and oeis.org/A061209 $\endgroup$ – JesseW Sep 21 '18 at 5:53
  • $\begingroup$ And the (non-finite, although I don't know if it has been proved) generalized sequence is oeis.org/A193163 $\endgroup$ – JesseW Sep 21 '18 at 5:56

The set $\{n\in\mathbb{N}\,|\,n\text{ and }n+1\text{ are perfect powers}\}$ has a largest element: $8$. Actually, it's its only element.

  • $\begingroup$ That's the kind of thing I was thinking of, thank you! $\endgroup$ – JesseW Sep 10 '18 at 6:30
  • $\begingroup$ The only reason I haven't accepted this answer yet is I'd ideally prefer an answer with multiple sets of varying sizes. $\endgroup$ – JesseW Sep 11 '18 at 15:58
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    $\begingroup$ @JesseW I understand that. Please don't feel pressured. $\endgroup$ – José Carlos Santos Sep 11 '18 at 16:46

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