2731
This number is a prime.
If 2731 people are chosen at random, then the probability that at least 17 of them share the same birthday is greater than 50%. By chance, 2731 is a Wagstaff prime, i.e., a prime of the form (2^p + 1)/3. Note that (2^17 + 1)/3 is also a Wagstaff prime.
A Wagstaff prime with a sum of digits equal to its exponent. [Barnes]
(There is one curio for this number that has not yet been approved by an editor.)
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