Most (Historically) "Distinguished" Number(s)?

[Rick Shepherd]

Recent messages remind me of some questions I posed to
myself recently:

What positive integer (or, ideally, sequence of numbers for the OEIS)
simultaneously possesses n properties, n maximal, where the
properties have all been considered important enough historically to
receive (well-known) names?

For example, prime Fibonacci numbers share two properties.
There are lots of similar examples with two shared properties
(already in the OEIS), involving, say, different types of
figurate numbers.

Recently I found that...
2,1597,?  are prime Fibonacci central polygonal numbers
(three shared properties).  1597 is also odd and lucky (5+
simultaneous properties), etc.  (I found no other prime
Fibonacci central polygonal numbers among the first
65000+ Fibonacci numbers.).  So, 1597 is a very first
shot at such a distinguished number, but...

How big does n, the number of properties, need to be in order
for the number or sequence to be truly "distinguished" or
remarkable?

Finally, because of the fact that "small" numbers are overworked
(and underpaid), I'm sure that any such distinguished numbers
must be at least of a certain size, whatever that may happen to mean.

Regards,
Rick Shepherd