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The Hardy-Ramanujan number is the smallest product of three distinct primes of the form 6n + 1. [Pol]
The largest number which is divisible by its prime sum of digits (19) and reversal (91)
happens to be Ramanujan's famous taxi-cab number
(1729 = 123 + 13 = 103 + 93).
It is the smallest number expressible as the sum of two positive cubes in two different ways.
The smallest number that is a pseudoprime simultaneously to bases 2, 3 and 5. [Pomerance ,
Selfridge and
Wagstaff]
If you reverse the middle digits of this pseudoprime you get 1279 and 21279 - 1 is a Mersenne prime. [Luhn]
Schiemann's first pair of isospectral lattices L+(1,7,13,19) and L-(1,7,13,19) are of determinant 1*7*13*19 = 1729. [Poo Sung]
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