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A072141
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Numbers n such that two applications of 'Reverse and Subtract' lead to n, whereas one application does not lead to n.
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14
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2178, 6534, 21978, 65934, 219978, 659934, 2199978, 6599934, 21782178, 21999978, 65346534, 65999934, 217802178, 219999978, 653406534, 659999934, 2178002178, 2197821978, 2199999978, 6534006534, 6593465934, 6599999934
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OFFSET
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1,1
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COMMENTS
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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).
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LINKS
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FORMULA
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n = f(f(n)), n <> f(n), where f: x -> |x - reverse(x)|.
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EXAMPLE
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6534 -> |6534 - 4356| = 2178 -> |2178 - 8712| = 6534.
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CROSSREFS
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KEYWORD
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base,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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