A072141 Numbers n such that two applications of 'Reverse and Subtract' lead to n, whereas one application does not lead to n.

2178, 6534, 21978, 65934, 219978, 659934, 2199978, 6599934, 21782178, 21999978, 65346534, 65999934, 217802178, 219999978, 653406534, 659999934, 2178002178, 2197821978, 2199999978, 6534006534, 6593465934, 6599999934

Offset

1,1

Comments

There are two four-digit 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.

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:

.k..smallest.solution..smallest.n.with.period.k..sequence

.1..................0.........................0.......---

.2...............2178......................1012..(this one)

14...........11436678..................10001145...A072142

22.......108811891188..............100000114412...A072143

12.......118722683079..............100010505595...A072718

17...1186781188132188..........1000000011011012...A072719

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.

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).

Links

Ray Chandler, Table of n, a(n) for n = 1..10000

Michael P. Greaney, 1012 and other such numbers, +plus magazine, August 30, 2017.

Formula

n = f(f(n)), n <> f(n), where f: x -> |x - reverse(x)|.

Example

6534 -> |6534 - 4356| = 2178 -> |2178 - 8712| = 6534.

Crossrefs

Cf. A072137, A072140, A072142, A072143, A073142, A073143, A073144.

Sequence in context: A250240 A157476 A157853 * A008918 A262792 A035770

Adjacent sequences: A072138 A072139 A072140 * A072142 A072143 A072144

Keyword

base,nonn

Author

Klaus Brockhaus, Jun 24 2002

Extensions

More terms from Ray Chandler, Oct 09 2017

Status

approved