

A023108


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


69



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
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 Reversethenadd function A056964 is repeatedly applied.  M. F. Hasler, Apr 13 2019
Empirical observation of the available list shows that every positive integer of the form 99*k2 such that 1<k<9 belongs to the sequence; every positive integer of the form 999*k1 such that 1<k<9 belongs to the sequence excepting k=5, which is a palindromic number; every positive integer of the form 9999*k such that 1<k<9 belongs to the sequence, and 99999*2+1 belongs to the sequence. Thus, it can be conjectured that every positive integer n in base 10 of the form 999...9*km, such that 1<k<9, m=4j, and j>1, where j counts the number of "9" composing the form of n, is either a palindromic number or belongs to the list.  Juan Moreno Borrallo, Aug 10 2020


REFERENCES

F. Gruenberger, Computer Recreations, Scientific American, 250 (No. 4, 1984), 1926.
Daniel Lignon, Dictionnaire de (presque) tous les nombres entiers, Ellipses, Paris, 2012, 702 pages. See Entry 196.


LINKS

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], 20122014.
Patrick De Geest, Some thematic websources
James Grime and Brady Haran, What's special about 196?, Numberphile video (2015).
R. K. Guy, What's left?, Math Horizons, Vol. 5, No. 4 (April 1998), pp. 57.
Tim Irvin, About Two Months of Computing, or An Addendum to Mr. Walker's Three Years of Computing
Project Euler, Problem 55: How many Lychrel numbers are there below tenthousand?
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!


EXAMPLE

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)


MATHEMATICA

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


PROG

(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


CROSSREFS

Cf. A006960, A088753, A063048, A089694, A089521, A023109; A075421, A030547.
Cf. A056964 ("reverse and add" operation on which this is based).
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



