Thanks again to Jen for finding this link.
The third cube, 33.
1/27 = .037037037..., and 1/37 = .027027027...
27!+1 is prime. There aren't many numbers N such that N!+1 is prime; and it's hard to find them because factorials are so large. As of May 2002, the only known values of N are: 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, 26951, ... (Sloane's A002981).
27×227+227+27 is prime.
Because 27 is the smallest factor of 999 that is not also a factor of 9 or 99, 27 is the smallest number whose reciprocal has a 3-digit repeating pattern. See also 239 and 757.
27 is the sum of the digits 2 through 7. 15 is the other 2-digit number that shares this property.
To test a number for divisibility by 27:
Take the digits of the number in groups of 3 starting from the right, and add the resulting numbers together. If the result is more than 3 digits, repeat this process.
Check the resulting 3-digit number for divisibility by 9. If it isn't, the original number isn't divisible by 27.
If it is, divide it by 9 (see the 9 entry for a simple way to do this). Then check the answer for divisibility by 3. If it is, the original number is divisible by 27, otherwise it isn't.
See also 89, 134217728, 10888869450418352160768000000 and 103.0056206947796095239×1029.
27 is a psychologically random number, similar to 17 and 37 and having no particular cultural origin. Like 37, it is often used when some random-sounding large number is needed. For example, in Graham Greene's 1953 play The Living Room one finds the line:
ROSE Since my last confession three weeks ago I've committed adultery
(27 is also my favorite cult number, for various reasons, for example it's the street number of a house where I grew up, and my age was 27 years + 27 days when I met a certain close friend.)