%I
%S 2178,6534,21978,65934,219978,659934,2199978,6599934,21782178,
%T 21999978,65346534,65999934,217802178,219999978,653406534,659999934,
%U 2178002178,2197821978,2199999978,6534006534,6593465934,6599999934
%N Numbers n such that two applications of 'Reverse and Subtract' lead to n, whereas one application does not lead to n.
%C There are two fourdigit terms in the sequence. Further terms are obtained (a) by inserting at the center of these terms any number of 9's and (b) by concatenating a term any number of times with itself and inserting an equal number of 0's at all junctures. Method (b) may be applied recursively to all terms.  Revised thanks to a comment from _Hans Havermann_, Jan 27 2004.
%C Solutions to x = f^k(x), x <> f^j(x) for j < k, where f: n > n  reverse(n), for period lengths k <= 22 are given by:
%C .k..smallest.solution..smallest.n.with.period.k..sequence
%C .1..................0.........................0.......
%C .2...............2178......................1012..(this one)
%C 14...........11436678..................10001145...A072142
%C 22.......108811891188..............100000114412...A072143
%C 12.......118722683079..............100010505595...A072718
%C 17...1186781188132188..........1000000011011012...A072719
%C I still have no answer to the question if there exist solutions for other values of k. Random tests for larger n (up to 50 digits) have shown that periods 1 and 2 are very frequent (> 90 %), period 14 is not unusual (7 to 8 %), periods 22, 12 and 17 are very rare and other periods did not appear.
%C I conjecture that for some k there are no solutions, while in other cases the minimal solutions will have 20, 24, 28, ... digits (which however are very hard to find).
%H Ray Chandler, <a href="/A072141/b072141.txt">Table of n, a(n) for n = 1..10000</a>
%H Michael P. Greaney, <a href="https://plus.maths.org/content/1012andothersuchnumbers">1012 and other such numbers</a>, +plus magazine, August 30, 2017.
%F n = f(f(n)), n <> f(n), where f: x > x  reverse(x).
%e 6534 > 6534  4356 = 2178 > 2178  8712 = 6534.
%Y Cf. A072137, A072140, A072142, A072143, A073142, A073143, A073144.
%K base,nonn
%O 1,1
%A _Klaus Brockhaus_, Jun 24 2002
%E More terms from _Ray Chandler_, Oct 09 2017
