Catch of the Day (153 Fishes)

The Bible tells of Jesus and the Apostles going fishing and catching 
exactly 153 fish.  It so happens that 153 is a "triangular" number 
(in the Pythagorean sense), being the sum of the first 17 integers, 
and it's also the sum of the first five factorials.  Also, 153 equals 
the sum of the cubes of its digits.  Moreover, if you take ANY 
integer divisible by 3 and add up the cubes of its digits, then take 
the result and sum the cubes of its digits, and so on, you invariably 
end up at 153.  For example, the number 4713 is a multiple of 3, so
we should be able to reach 153 by iteratively summing the cubes of
the digits.  Let's have a look:

              starting number        = 4713
              4^3 + 7^3 + 1^3 + 3^3  =  435
              4^4 + 3^3 + 5^3        =  216
              2^3 + 1^3 + 6^3        =  225
              2^3 + 2^3 + 5^3        =  141
              1^3 + 4^3 + 1^3        =   66
              6^3 + 6^3              =  432
              4^3 + 3^3 + 2^3        =   99
              9^3 + 9^3              = 1458
              1^3 + 4^3 + 5^3 + 8^3  =  702
              7^3 + 2^3              =  351
              3^3 + 5^3 + 1^3        =  153  (whew)

One wonders how much, if any, of this was known to the author of 
the Gospel.  Since our modern decimal number system wasn't officially
invented until much later, it might seem implausible that the number
153 was selected on the basis of any properties of its decimal digits.
On the other hand, the text does specifically state the number 
verbally in explicit decimal form, i.e., "Simon Peter went up, and
drew the net to land full of great fishes, an hundred and fifty
and three: and for all there was so many, yet was not the net broken."
(John, 21:11)  So rather than talking about scores or dozens, it
speaks in multiples of 100, 10, and 1.

Of course, we could perform a similar iteration on the digits of
a number in any base.  One of the more interesting cases is the
base 14, in which 2/3 of all number eventually fall into a particular
cycle.  Coincidentally, this cycle includes the decimal number 153,
but it also includes 26 other numbers, for a total length of 27,
which is 3 cubed (which the mystically minded should have no trouble
associating with the Trinity).  The decimal values of this base-14 
cycle are

     9   729  1028   368  1793   738  2027  2395  1756  
  2925  3926   433  2213  1396  1344  1944  4185  2605
  2262  2186  1347  1971  2331  3402   153  3197   198

It's interesting that although the numbers that reduce to 153 in
the decimal version of this iteration are easily characterized as
precisely those numbers that are divisible by 3, in the case of
this base-14 iteration it's much more difficult to see a consistent
pattern in the numbers that reduce to the above cycle, even though
they evidently constitute 2/3 of all numbers.