196-Algorithm

Take any Positive Integer of two Digits or more, reverse the Digits, and add to the original number. Now repeat the procedure with the Sum so obtained. This procedure quickly produces Palindromic Numbers for most Integers. For example, starting with the number 5280 produces (5280, 6105, 11121, 23232). The end results of applying the algorithm to 1, 2, 3, ... are 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 11, 33, 44, 55, 66, 77, 88, 99, 121, ... (Sloane's A033865). The value for 89 is especially large, being 8813200023188.

The first few numbers not known to produce Palindromes are 196, 887, 1675, 7436, 13783, ... (Sloane's A006960), which are simply the numbers obtained by iteratively applying the algorithm to the number 196. This number therefore lends itself to the name of the Algorithm.

The number of terms $a(n)$ in the iteration sequence required to produce a Palindromic Number from $n$ (i.e., $a(n)=1$ for a Palindromic Number, $a(n)=2$ if a Palindromic Number is produced after a single iteration of the 196-algorithm, etc.) for $n=1$, 2, ... are 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 1, ... (Sloane's A030547). The smallest numbers which require $n=0$, 1, 2, ... iterations to reach a palindrome are 0, 10, 19, 59, 69, 166, 79, 188, ... (Sloane's A023109).

See also Additive Persistence, Digitaddition, Multiplicative Persistence, Palindromic Number, Palindromic Number Conjecture, RATS Sequence, Recurring Digital Invariant

References

Gardner, M. Mathematical Circus: More Puzzles, Games, Paradoxes and Other Mathematical Entertainments from Scientific American. New York: Knopf, pp. 242-245, 1979.

Gruenberger, F. ``How to Handle Numbers with Thousands of Digits, and Why One Might Want to.'' Sci. Amer. 250, 19-26, Apr. 1984.

Sloane, N. J. A. Sequences A023109, A030547, A033865, and A006960/M5410 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.