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A072137
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Length of the preperiodic part of the 'Reverse and Subtract' trajectory of n.
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21
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0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 6, 4, 5, 3, 3, 5, 4, 6, 2, 1, 2, 6, 4, 5, 3, 3, 5, 4, 6, 2, 1, 2, 6, 4, 5, 3, 3, 5, 4, 6, 2, 1, 2, 6, 4, 5, 3, 3, 5, 4, 6, 2, 1, 2, 6, 4, 5, 3, 3, 5, 4, 6, 2, 1, 2, 6, 4, 5, 3, 3, 5, 4, 6, 2, 1, 2, 6, 4, 5, 3, 3, 5, 4, 6, 2, 1, 2, 6, 4, 5, 3, 3, 5, 4, 6, 2, 1, 2, 1, 2, 6, 4
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OFFSET
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0,11
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COMMENTS
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'Reverse and Subtract' (cf. A070837, A070838) is defined by x -> |x - reverse(x)|, where reverse(x) is the digit reversal of x.
For every n the trajectory eventually becomes periodic, since 'Reverse and Subtract' does not increase the number of digits and so the set of available terms is finite. For small n the period length is 1, the periodic part consists of 0's, the last term of the preperiodic part is a palindrome.
The first n with period length 2 and a nontrivial periodic part is 1012 (cf. A072140).
This sequence is a weak analog of A033665, which uses 'Reverse and Add'.
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LINKS
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EXAMPLE
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a(15) = 4 since 15 -> |15- 51| = 36 -> |36 - 63| = 27 -> |27 - 72| = 45 -> |45 - 54| = 9.
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MATHEMATICA
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a[n_] := (k = 0; FixedPoint[ (k++; Abs[# - FromDigits[ Reverse[ IntegerDigits[#] ] ] ]) &, n]; k - 1); Table[ a[n], {n, 0, 104}] (* Jean-François Alcover, Dec 01 2011 *)
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PROG
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(Haskell)
import Data.List(inits, find); import Data.Maybe(fromJust)
a072137 :: Int -> Int
a072137 = length . fst . spanCycle (abs . a056965) where
spanCycle :: Eq a => (a -> a) -> a -> ([a], [a])
spanCycle f x = fromJust $ find (not . null . snd) $
zipWith (span . (/=)) xs $ inits xs
where xs = iterate f x
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CROSSREFS
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KEYWORD
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base,easy,nonn,nice
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AUTHOR
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STATUS
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approved
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