

A023108


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


65



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
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OFFSET

1,1


COMMENTS

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, 10digit numbers have 49.61% of Lychrels. So beyond 10 digits, Lychrels start to outnumber nonLychrels.  Dmitry Kamenetsky, Oct 12 2015


REFERENCES

F. Gruenberger, Computer Recreations, Scientific American, 250 (No. 4, 1984), 1926.
R. K. Guy, What's left?, preprint, 1998.
Daniel Lignon, Dictionnaire de (presque) tous les nombres entiers, Ellipses, Paris, 2012, 702 pages. See Entry 196.


LINKS

William Boyles, Table of n, a(n) for n = 1..249 (Terms 1...45 from David W. Wilson)
Jason Doucette, World Records
Martianus Frederic Ezerman, Bertrand Meyer and Patrick Sole, On Polynomial Pairs of Integers, arXiv:1210.7593 [math.NT], 20122014.
P. De Geest, Some thematic websources
James Grime and Brady Haran, What's special about 196?, Numberphile video (2015).
T. Irvin, About Two Months of Computing, or An Addendum to Mr. Walker's Three Years of Computing
Wade VanLandingham, 196 and other Lychrel numbers
Project Euler, Problem 55: How many Lychrel numbers are there below tenthousand?
Wade VanLandingham, Largest known Lychrel number
J. 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!


MATHEMATICA

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


CROSSREFS

Cf. A006960, A088753, A063048, A089694, A089521, A023109. A075421, A030547.
Sequence in context: A224667 A118781 A119667 * A092231 A188247 A211851
Adjacent sequences: A023105 A023106 A023107 * A023109 A023110 A023111


KEYWORD

nonn,base,nice


AUTHOR

David W. Wilson


EXTENSIONS

Edited by M. F. Hasler, Dec 04 2007


STATUS

approved



