From: kfl@KeithLynch.net   
      
   Joy Beeson  wrote:   
   > While counting my calf raises tonight, it struck me that two to the   
   > third was followed immediately by three to the second.   
      
   > I wonder what pairs like this are called?   
      
   Perfect powers.  Catalan's Conjecture -- which has since been   
   proven -- is that 8 and 9 are the only adjacent ones.  See   
   https://oeis.org/A001597   
      
   Note that 2025 is a perfect power, the first since 1936 and the last   
   until 2048.   
      
   > I've long been aware that sixteen is its own whatsit:  it's both   
   > four to the second and two to the fourth.  Sixteen must be the only   
   > such number, aside from numbers are a number raised to itself.   
      
   Any perfect power whose exponent isn't prime is also the perfect power   
   with every of exponent of that exponent's divisors.  For instance 64   
   is not just 2^6 but also 4^3 and 8^2, since 6=3*2.   
      
   Once when someone asked me how old I was, I told them that my age in   
   days that week was both a square and a cube.  That meant that my age   
   was also a sixth power, and there was only one plausible age in days   
   that was a sixth power for me, since I'm obviously neither a child nor   
   the oldest person who ever lived.   
      
   > I'll bet that those numbers also have a name.  The series   
   > 1^1, 2^2, . . . has a rising rate that puts factorials to   
   > shame.   
      
   Apparently not, other than n^n, but you can read all about it at   
   https://oeis.org/A000312   
      
   oeis.org is the Online Encyclopedia of Integer Sequences, founded 61   
   years ago, and still run by its founder.  You can search within it for   
   my name to read about my humble contributions to it.   
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   Keith F. Lynch - http://keithlynch.net/   
   Please see http://keithlynch.net/email.html before emailing me.   
      
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