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

Author

Klaus Brockhaus, Jun 24 2002

Keywords

• base
• easy
• nonn
• nice

Comments

'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'.

Examples

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

Links

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

Crossrefs

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

Programs

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
-- Reinhard Zumkeller, Oct 24 2010

Mathematica

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 *)