A089521 -id:A089521 - cp's OEIS frontend

A089522 a(n) = smallest non-palindromic number k such that the Reverse and Add! trajectory of k joins the trajectory of A089521(n).

19098, 199998, 990999, 1909098, 19002998, 9905999, 11009094, 19003098, 19005098, 19007098, 19999998, 190009098, 990959999, 990969999, 1990029998, 1990049998, 1990069998, 1990089998, 999079999, 1999049998, 1999069998

Offset: 1

Klaus Brockhaus, Nov 10 2003

Terms of A063048 that are not terms of A088753 (not in ascending order). a(n) > A088753(n). a(n) is the substitute so to speak of A088753(n) in A063048.

a(12), a(15) to a(18), a(20), a(21) are conjectural; it is not yet ensured that they are minimal.

			The trajectories of A088753(1) = 9999 and of 19098 join at 2089791 and there is no number between 9999 and 19098 whose trajectory joins that of 9999.

Index entries for sequences related to Reverse and Add!

Cf. A063048, A088753, A089521.

A023108 Positive integers which apparently never result in a palindrome under repeated applications of the function A056964(x) = x + (x with digits reversed).

Original entry on oeis.org

196, 295, 394, 493, 592, 689, 691, 788, 790, 879, 887, 978, 986, 1495, 1497, 1585, 1587, 1675, 1677, 1765, 1767, 1855, 1857, 1945, 1947, 1997, 2494, 2496, 2584, 2586, 2674, 2676, 2764, 2766, 2854, 2856, 2944, 2946, 2996, 3493, 3495, 3583, 3585, 3673, 3675

Offset: 1

David W. Wilson

196 is conjectured to be smallest initial term which does not lead to a palindrome. John Walker, Tim Irvin and others have extended this to millions of digits without finding one (see A006960).
Also called Lychrel numbers, though the definition of "Lychrel number" varies: Purists only call the "seeds" or "root numbers" Lychrel; the "related" or "extra" numbers (arising in the former's orbit) have been coined "Kin numbers" by Koji Yamashita. There are only 2 "root" Lychrels below 1000 and 3 more below 10000, cf. A088753. - M. F. Hasler, Dec 04 2007
Question: when do numbers in this sequence start to outnumber numbers that are not in the sequence? - J. Lowell, May 15 2014
Answer: according to Doucette's site, 10-digit numbers have 49.61% of Lychrels. So beyond 10 digits, Lychrels start to outnumber non-Lychrels. - Dmitry Kamenetsky, Oct 12 2015
From the current definition it is unclear whether palindromes are excluded from this sequence, cf. A088753 vs A063048. 9999 would be the first palindromic term that will never result in a palindrome when the Reverse-then-add function A056964 is repeatedly applied. - M. F. Hasler, Apr 13 2019

			From _M. F. Hasler_, Feb 16 2020: (Start)
Under the "add reverse" operation, we have:
196 (+ 691) -> 887 (+ 788) -> 1675 (+ 5761) -> 7436 (+ 6347) -> 13783 (+ 38731) -> etc. which apparently never leads to a palindrome.
Similar for 295 (+ 592) -> 887, 394 (+ 493) -> 887, 790 (+ 097) -> 887 and 689 (+ 986) -> 1675, which all merge immediately into the above sequence, and also for the reverse of any of the numbers occurring in these sequences: 493, 592, 691, 788, ...
879 (+ 978) -> 1857 -> 9438 -> 17787 -> 96558 is the only other "root" Lychrel below 1000 which yields a sequence distinct from that of 196. (End)

Daniel Lignon, Dictionnaire de (presque) tous les nombres entiers, Ellipses, Paris, 2012, 702 pages. See Entry 196.

A.H.M. Smeets, Table of n, a(n) for n = 1..20000 (tested for 200 iterations; first 249 terms from William Boyles)
DeCode, Lychrel Number, dCode.fr 'toolkit' to solve games, riddles, geocaches, 2020.
Jason Doucette, World Records
Martianus Frederic Ezerman, Bertrand Meyer and Patrick Sole, On Polynomial Pairs of Integers, arXiv:1210.7593 [math.NT], 2012-2014.
Patrick De Geest, Some thematic websources
James Grime and Brady Haran, What's special about 196?, Numberphile video (2015).
Fred Gruenberger, How to handle numbers with thousands of digits, and why one might want to, Computer Recreations, Scientific American, 250 (No. 4, 1984), 19-26.
R. K. Guy, What's left?, Math Horizons, Vol. 5, No. 4 (April 1998), pp. 5-7.
Tim Irvin, About Two Months of Computing, or An Addendum to Mr. Walker's Three Years of Computing
Niphawan Phoopha and Prapanpong Pongsriiam, Notes on 1089 and a Variation of the Kaprekar Operator, Int'l J. Math. Comp. Sci. (2021) Vol. 16, No. 4, 1599-1606.
Project Euler, Problem 55: How many Lychrel numbers are there below ten-thousand?
A.H.M. Smeets, Distribution of terms < 10000000 (number of terms in interval of length 10000)
Wade VanLandingham, 196 and other Lychrel numbers
Wade VanLandingham, Largest known Lychrel number
John Walker, Three Years Of Computing: Final Report On The Palindrome Quest
Eric Weisstein's World of Mathematics, 196 Algorithm.
Eric Weisstein's World of Mathematics, Palindromic Number Conjecture
Eric Weisstein's World of Mathematics, Lychrel Number
Index entries for sequences related to Reverse and Add!

Cf. A006960, A088753, A063048, A089694, A089521, A023109; A075421, A030547.

Cf. A056964 ("reverse and add" operation on which this is based).

With[{lim = 10^3}, Select[Range@ 4000, Length@ NestWhileList[# + IntegerReverse@ # &, #, ! PalindromeQ@ # &, 1, lim] == lim + 1 &]] (* Michael De Vlieger, Dec 23 2017 *)

select( {is_A023108(n, L=exponent(n+1)*5)=while(L--&& n*2!=n+=A004086(n),);!L}, [1..3999]) \\ with {A004086(n)=fromdigits(Vecrev(digits(n)))}; default value for search limit L chosen according to known records A065199 and indices A065198. - M. F. Hasler, Apr 13 2019, edited Feb 16 2020

Edited by M. F. Hasler, Dec 04 2007

cp's OEIS Frontend

A089522 a(n) = smallest non-palindromic number k such that the Reverse and Add! trajectory of k joins the trajectory of A089521(n).

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A023108 Positive integers which apparently never result in a palindrome under repeated applications of the function A056964(x) = x + (x with digits reversed).

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A089694 Number of palindromes < 10^n in A088753.

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