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Google's second challange is a bit more... vague.

f(1)= 7182818284 
f(2)= 8182845904 
f(3)= 8747135266 
f(4)= 7427466391 
f(5)= __________ 

The proper number in this sequence will grant you access to the next stage of 
game.  Sequences, I hate's them I do.  Hates them.

And does anyone get the feeling Google has a thing for 10 digit numbers?

So, the easy options are an additive, multiplicative, etc style sequence, but
that doesn't work.  If these are a commonly known sequence to mathemeticians
it's certainly not known to me.  These could describe points in a curve, but
I'm missing enough info to recreate the curve.

These numbers all have SOMETHING in common, I just don't see what off the top
of my head.  I'll just have to use my fallback, Google to get some suggestions.

Googling 'number sequence' gets me Number Cracker.  Play time is always 
helpful...  My brain hurts more.

Further googling turned up some math for various sequences, such as the
Fibonacci sequence.

I then hit this gem, the On-Line Encyclopedia of Integer Sequences.  It 
includes a search function.

Punch in the numbers, and we get this entry.  Series of 10 digit 
numbers whose units added up total 49, from the digits of e.  Surprise 
surprise, e again.  I'm picking up on a theme again.  So, no perl code 
required this time, just some good old searching.