A072138 -id:A072138 - cp's OEIS frontend

A072137 Length of the preperiodic part of the 'Reverse and Subtract' trajectory of n.

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

Offset: 0

Klaus Brockhaus, Jun 24 2002

'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'.

			a(15) = 4 since 15 -> |15- 51| = 36 -> |36 - 63| = 27 -> |27 - 72| = 45 -> |45 - 54| = 9.

R. Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A033665, A070837, A070838, A072138, A072139, A072140, A072141, A072146, A072147.

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
-- Reinhard Zumkeller, Oct 24 2010

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 *)

A109891 Least number that requires n steps to reach 0 by repeated application of f: x -> abs(x - reverse(x)).

Original entry on oeis.org

0, 1, 10, 16, 14, 15, 13, 1011, 1017, 1037, 1027, 1014, 1013, 1028, 100143, 100135, 100145, 100134, 100103, 100195, 100137, 100227, 100114, 100128, 100194, 100107, 100307, 100277, 100413, 100345, 100429, 100215, 100444, 100237, 100433, 100335

Offset: 0

Amarnath Murthy, Jul 13 2005

Coincides with A072138 for the first 14 and many later terms.

			f(16) = 61-16 = 45, f(45) = 54-45 = 9, f(9) = 9-9 = 0. For no k < 16 exactly three steps lead to 0, hence a(3) = 16.

Cf. A072137, A072138, A072141.

Edited, corrected and extended by Klaus Brockhaus, Jul 14 2005

cp's OEIS Frontend

A072137 Length of the preperiodic part of the 'Reverse and Subtract' trajectory of n.

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