Math Newsletter number 23; Wednesday, December 29, 2010.
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What's special about integers 11 through 20?

http://www.daviddarling.info/encyclopedia/N/numbers_types.html

11 is the smallest repunit prime.
http://en.wikipedia.org/wiki/Repunit_prime#Repunit_primes

12 is the smallest abundant number.
http://www.daviddarling.info/encyclopedia/A/abundant_number.html

13 is the number of Archimedian solids.
http://en.wikipedia.org/wiki/Archimedean_solid

14 is the smallest number n with the property that there are
no numbers relatively prime to n smaller numbers.
Proof: The number of smaller positive integers relative prime
to z = p1**k1 p2**k2 p3**k3 ...
is p1 p1**(k1-1) p2 p2**(k2-1) p3 p3**(k3-1) ...
Suppose 14 is the number of numbers relatively prime to z.
14 =
(p1-1) p1**(k1-1) (p2-1) p2**(k2-1) (p3-1) p3**(k3-1) ....
14 = 2 * 7
p1 = 2+1 = 3
p2 = ?; (7+1) is not prime.
14 is not of the form
(p1-1) p1**(k1-1) (p2-1) p2**(k2-1) (p3-1) p3**(k3-1) ....

15 is the smallest composite number n with the property that
there is only one group of order n.
http://en.wikipedia.org/wiki/List_of_small_groups

16 is the only integer of the form x**y = y**x with x and y
different integers.
Proof by hand waving and inspection. :)

17 is the number of wallpaper groups.
http://en.wikipedia.org/wiki/Wallpaper_group

18 is the smallest difference between an emirp and its
reverse. 18 = 31 - 13.
http://en.wikipedia.org/wiki/Emirp

19 is the maximum number of 4th powers needed to sum to any
number.
http://primes.utm.edu/curios/page.php/19.html

19 is the only prime which is equal to the difference of two
cubes of primes; 19 = 3**3 - 2**3.

The Rhind papyrus contained a problem to find x so that x
plus one seventh of x will equal 19.
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