A072137 - OEIS

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

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OFFSET

0,11

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'.

LINKS

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

EXAMPLE

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

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 *)

PROG

(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

CROSSREFS

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