4001

This number is a prime.

The only prime less than a googol of form p=(x+1)*10^x+1, for x a prime, (case x=3). [Loungrides]

The total number of 1's used in writing out (in figures) all of the first ten thousand (10000) numbers. [Beedassy]

The smallest prime number whose square (16008001) contains exactly 50% zeros. Note that the prime itself is also exactly 50% zeros. [Gaydos]