cp's OEIS Frontend

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

%I A023108 #155 Feb 16 2025 08:32:34
%S A023108 196,295,394,493,592,689,691,788,790,879,887,978,986,1495,1497,1585,
%T A023108 1587,1675,1677,1765,1767,1855,1857,1945,1947,1997,2494,2496,2584,
%U A023108 2586,2674,2676,2764,2766,2854,2856,2944,2946,2996,3493,3495,3583,3585,3673,3675
%N A023108 Positive integers which apparently never result in a palindrome under repeated applications of the function A056964(x) = x + (x with digits reversed).
%C A023108 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).
%C A023108 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
%C A023108 Question: when do numbers in this sequence start to outnumber numbers that are not in the sequence? - _J. Lowell_, May 15 2014
%C A023108 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
%C A023108 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
%D A023108 Daniel Lignon, Dictionnaire de (presque) tous les nombres entiers, Ellipses, Paris, 2012, 702 pages. See Entry 196.
%H A023108 A.H.M. Smeets, <a href="/A023108/b023108.txt">Table of n, a(n) for n = 1..20000</a> (tested for 200 iterations; first 249 terms from William Boyles)
%H A023108 DeCode, <a href="https://www.dcode.fr/lychrel-number">Lychrel Number</a>, dCode.fr 'toolkit' to solve games, riddles, geocaches, 2020.
%H A023108 Jason Doucette, <a href="http://www.jasondoucette.com/worldrecords.html">World Records</a>
%H A023108 Martianus Frederic Ezerman, Bertrand Meyer and Patrick Sole, <a href="https://arxiv.org/abs/1210.7593">On Polynomial Pairs of Integers</a>, arXiv:1210.7593 [math.NT], 2012-2014.
%H A023108 Patrick De Geest, <a href="http://www.worldofnumbers.com/weblinks.htm">Some thematic websources</a>
%H A023108 James Grime and Brady Haran, <a href="https://www.youtube.com/watch?v=bN8PE3eljdA">What's special about 196?</a>, Numberphile video (2015).
%H A023108 Fred Gruenberger, <a href="https://www.jstor.org/stable/24969338">How to handle numbers with thousands of digits, and why one might want to</a>, Computer Recreations, Scientific American, 250 (No. 4, 1984), 19-26.
%H A023108 R. K. Guy, <a href="https://www.jstor.org/stable/25678158">What's left?</a>, Math Horizons, Vol. 5, No. 4 (April 1998), pp. 5-7.
%H A023108 Tim Irvin, <a href="http://www.fourmilab.ch/documents/threeyears/two_months_more.html">About Two Months of Computing, or An Addendum to Mr. Walker's Three Years of Computing</a>
%H A023108 Niphawan Phoopha and Prapanpong Pongsriiam, <a href="http://ijmcs.future-in-tech.net/16.4/R-Phoopha-Pongsriiam.pdf">Notes on 1089 and a Variation of the Kaprekar Operator</a>, Int'l J. Math. Comp. Sci. (2021) Vol. 16, No. 4, 1599-1606.
%H A023108 Project Euler, <a href="https://projecteuler.net/problem=55">Problem 55: How many Lychrel numbers are there below ten-thousand?</a>
%H A023108 A.H.M. Smeets, <a href="/A023108/a023108.png">Distribution of terms < 10000000 (number of terms in interval of length 10000)</a>
%H A023108 Wade VanLandingham, <a href="http://www.p196.org/">196 and other Lychrel numbers</a>
%H A023108 Wade VanLandingham, <a href="http://web.archive.org/web/20030828235934/http://home.cfl.rr.com/p196/lychrel.html">Largest known Lychrel number</a>
%H A023108 John Walker, <a href="http://www.fourmilab.ch/documents/threeyears/threeyears.html">Three Years Of Computing: Final Report On The Palindrome Quest</a>
%H A023108 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/196-Algorithm.html">196 Algorithm.</a>
%H A023108 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PalindromicNumberConjecture.html">Palindromic Number Conjecture</a>
%H A023108 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LychrelNumber.html">Lychrel Number</a>
%H A023108 <a href="/index/Res#RAA">Index entries for sequences related to Reverse and Add!</a>
%e A023108 From _M. F. Hasler_, Feb 16 2020: (Start)
%e A023108 Under the "add reverse" operation, we have:
%e A023108 196 (+ 691) -> 887 (+ 788) -> 1675 (+ 5761) -> 7436 (+ 6347) -> 13783 (+ 38731) -> etc. which apparently never leads to a palindrome.
%e A023108 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, ...
%e A023108 879 (+ 978) -> 1857 -> 9438 -> 17787 -> 96558 is the only other "root" Lychrel below 1000 which yields a sequence distinct from that of 196. (End)
%t A023108 With[{lim = 10^3}, Select[Range@ 4000, Length@ NestWhileList[# + IntegerReverse@ # &, #, ! PalindromeQ@ # &, 1, lim] == lim + 1 &]] (* _Michael De Vlieger_, Dec 23 2017 *)
%o A023108 (PARI) 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
%Y A023108 Cf. A006960, A088753, A063048, A089694, A089521, A023109; A075421, A030547.
%Y A023108 Cf. A056964 ("reverse and add" operation on which this is based).
%K A023108 nonn,base,nice
%O A023108 1,1
%A A023108 _David W. Wilson_
%E A023108 Edited by _M. F. Hasler_, Dec 04 2007