A072146 -id:A072146 - 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 *)

A072138 Smallest k whose 'Reverse and Subtract' trajectory has a preperiodic part of length n.

Original entry on oeis.org

0, 1, 10, 16, 14, 15, 13, 1011, 1017, 1037, 1027, 1014, 1013, 1028, 100113, 100104, 100145, 100134, 100103, 100112, 100133, 100187, 100114, 100128, 100194, 100107, 100307, 100277, 100413, 100345, 100429, 100215, 100427, 100214, 100433, 100335

Offset: 0

Klaus Brockhaus, Jun 24 2002

'Reverse and Subtract' (cf. A072137) is defined by x -> |x - reverse(x)|. For small n the last term of the preperiodic part of the trajectory (cf. A072139) is a palindrome, so this sequence is a weak analog of A033665, which uses 'Reverse and Add'. - 1012 is the first n such that last term of the preperiodic part is not palindromic (cf. A072140).

			a(8) = 1017, since 1017 is the smallest number whose 'Reverse and Subtract' trajectory has eight preperiodic terms: 1017 -> 6084 -> 1278 -> 7443 -> 3996 -> 2997 -> 4995 -> 999.

Cf. A033665, A072137, A072139, A072140, A072146.

A072147 Records for the length of the preperiodic part of the 'Reverse and Subtract' trajectories.

Original entry on oeis.org

0, 1, 2, 6, 7, 12, 13, 18, 25, 40, 45, 47, 48, 49, 55, 56, 60, 62, 63, 64, 66, 71, 72, 75, 78, 81, 106, 108, 111, 112, 114, 115, 119, 121, 122, 130, 132, 133, 135, 147, 148, 149, 151, 156

Offset: 1

Klaus Brockhaus, Jun 24 2002

Successive maxima in sequence A072137. A072146 gives the corresponding starting points. - This sequence is a weak analog of A065199, which uses 'Reverse and Add'.

			6 is a record, since the preperiodic part of the trajectory of 13 has length 6 and for k < 13 the preperiodic part has a smaller length (at most 2).

Cf. A072137, A072146, A065199.

a(18) inserted, a(25) corrected, a(29) through a(44) added by Alexander Pesch (alex-physics(AT)gmx.net), May 29 2007

Edited by N. J. A. Sloane, Dec 01 2007

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

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A072138 Smallest k whose 'Reverse and Subtract' trajectory has a preperiodic part of length n.

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A072147 Records for the length of the preperiodic part of the 'Reverse and Subtract' trajectories.

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