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ITEM 63 (Schroeppel, etc.):
The joys of 239 are as follows:
* pi = 16 arctan (1/5) - 4 arctan(1/239),
* which is related to the fact that 2 * 13^4 - 1 = 239^2,
* which is why 239/169 is an approximant (the 7th) of sqrt(2).
* arctan(1/239) = arctan(1/70) - arctan(1/99) = arctan(1/408) +
arctan(1/577)
* 239 needs 4 squares (the maximum) to express it.
* 239 needs 9 cubes (the maximum, shared only with 23) to express it.
* 239 needs 19 fourth powers (the maximum) to express it.
* (Although 239 doesn't need the maximum number of fifth powers.)
* 1/239 = .00418410041841..., which is related to the fact that
* 1,111,111 = 239 * 4,649.
* 239 = 11101111 base 2.
* 239 = 22212 base 3.
* 239 = 3233 base 4.
* There are 239 primes < 1500.
* K239 is Mozart's only work for 2 orchestras.
* Guess what memo this is.
* And 239 is prime, of course.
Added by Alex Glushkovsky:
239 = 16^2-16^1-16^0
where 16 = (2-1)(3-1)(9-1)
also:
239 = 2^8-2^4-2^0 = 2^(2x2x2)-2^(2x2)-2^(2-2)