A023109 -id:A023109 - 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).

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

A006960 Reverse and Add! sequence starting with 196.

Original entry on oeis.org

196, 887, 1675, 7436, 13783, 52514, 94039, 187088, 1067869, 10755470, 18211171, 35322452, 60744805, 111589511, 227574622, 454050344, 897100798, 1794102596, 8746117567, 16403234045, 70446464506, 130992928913, 450822227944, 900544455998, 1800098901007, 8801197801088, 17602285712176

Offset: 0

N. J. A. Sloane, Simon Plouffe

196 is conjectured to be the smallest initial term which does not lead to a palindrome. John Walker, Tim Irvin and others have extended the trajectory of 196 to millions of digits without finding a palindrome.
From A.H.M. Smeets, Jan 31 2019: (Start)
Palindromes for a(9)/2, a(14)/2 and a(20)/2.
Observed: It seems that most, but not all, Lychrel numbers (seeds given in A063048) have a trajectory term that, divided by 2, becomes palindromic. Note that 196 is the first Lychrel number (A063048(1)). (End)
Observed: On average, 0.414 digits are gained by each step of the reverse and add procedure; i.e., 2.416 steps are needed on average to gain a factor of 10. This holds for any trajectory of reverse and add for decimal number representation. - A.H.M. Smeets, Feb 03 2019

			From _M. F. Hasler_, Apr 13 2019: (Start)
Start with 196 = a(0), then:
A056964(196) = 196 + 691 = 887 = a(1); then:
A056964(887) = 887 + 788 = 1675 = a(2); then:
A056964(1675) = 1675 + 5761 = 7436 = a(3); then:
A056964(7436) = 7436 + 6347 = 13783 = a(4); then:
A056964(13783) = 13783 + 38731 = 52514 = a(5); etc. (End)

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 196, p. 58, Ellipses, Paris 2008.
D. H. Lehmer, "Sujets d'étude. No. 74," Sphinx (Bruxelles), 8 (1938), 12-13. (This is the currently earliest known reference to the 196 Problem). - James D. Klein, Apr 09 2012
Daniel Lignon, Dictionnaire de (presque) tous les nombres entiers, Ellipses, Paris, 2012, 702 pages. See Entry 196.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 70.
Popular Computing (Calabasas, CA), The 196 Problem, Vol. 3 (No. 30, Sep 1975), pages PC30-6 to PC30-9.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe and Michael Lee, Table of n, a(n) for n = 0..2390 (T. D. Noe supplied terms 0 to 200)
Patrick De Geest, Some thematic websources
Jason Doucette, World Records
Martianus Frederic Ezerman, Bertrand Meyer and Patrick Solé, On Polynomial Pairs of Integers, arXiv:1210.7593 [math.NT], 2012. - From _N. J. A. Sloane_, Nov 08 2012
Felix Fröhlich, C++ program for this sequence
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
Madras Math's Amazing Number Facts, The Ultimate Palindrome
I. Peter, More trajectories
Wade VanLandingham, 196 and Other Lychrel Numbers
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.
Index entries for sequences related to Reverse and Add!

Cf. A023108, A023109, A033665, A016016, A056964, A004086.

a006960 n = a006960_list !! n
a006960_list = iterate a056964 196 -- Reinhard Zumkeller, Sep 22 2011

a:= proc(n) option remember; `if`(n=0, 196, (h-> h+ (s->
      parse(cat(s[-i]$i=1..length(s))))(""||h))(a(n-1)))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 25 2014

a = {196}; For[i = 2, i < 26, i++, a = Append[a, a[[i - 1]] + ToExpression[ StringReverse[ToString[a[[i - 1]]]]]]]; a
NestList[#+FromDigits[Reverse[IntegerDigits[#]]]&,196,25] (* Harvey P. Dale, Jun 05 2011 *)
NestList[#+IntegerReverse[#]&,196,25] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 04 2019 *)

A006960_vec(N=99)=vector(N,i,N=if(i>1,A056964(N),196)) \\ M. F. Hasler, Apr 13 2019

a(n+1) = A056964(a(n)). - A.H.M. Smeets, Jan 27 2019

More terms from Vit Planocka (planocka(AT)mistral.cz), Sep 28 2002

A033865 Start with n; if palindrome, stop; otherwise add to itself with digits reversed; a(n) gives palindrome at which it stops, or -1 if no palindrome is ever reached.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 11, 33, 44, 55, 66, 77, 88, 99, 121, 22, 33, 22, 55, 66, 77, 88, 99, 121, 121, 33, 44, 55, 33, 77, 88, 99, 121, 121, 363, 44, 55, 66, 77, 44, 99, 121, 121, 363, 484, 55, 66, 77, 88, 99, 55, 121, 363, 484, 1111, 66, 77, 88, 99, 121, 121, 66, 484, 1111, 4884, 77, 88, 99, 121, 121, 363, 484, 77, 4884, 44044, 88

Offset: 0

David W. Wilson

It is believed that a(196) = -1.

			19 -> 19 + 91 = 110 -> 110 + 011 = 121, so a(19) = 121.

M. Donner, I Love Me, Vol. I: S. Wordrow's palindromic encyclopedia (Algonquin Books, 1996) p. 268

T. D. Noe, Table of n, a(n) for n = 0..195
O. Forster, ARIBAS
Mathforum, Making Numbers into Palindromic Numbers
Eric Weisstein's World of Mathematics, 196-Algorithm.

Cf. A061563, A016016, A023109, A006960, A023108, A002113, A033665 (number of steps).

var st: stack; end; for k := 0 to 60 do n := k; while n <> int_reverse(n) do n := n + int_reverse(n); end; stack_push(st,n); end; stack2array(st);

Table[NestWhile[# + FromDigits[Reverse[IntegerDigits[#]]] &, n, IntegerDigits[#] != Reverse[IntegerDigits[#]] &], {n, 0, 90}] (* Harvey P. Dale, Dec 18 2011 *)

a(n)=my(k); while((k=fromdigits(Vecrev(digits(n)))) != n, n += k); n \\ infinite loop if a(n) = -1; Charles R Greathouse IV, Dec 13 2015

More terms from Jenise Smalley (neicey01(AT)hotmail.com), Oct 18 2001

A065198 Indices of record high values in A033665, ignoring those numbers that are believed never to reach a palindrome.

Original entry on oeis.org

0, 10, 19, 59, 69, 79, 89, 10548, 10677, 10833, 10911, 147996, 150296, 1000689, 1005744, 1007601, 7008899, 9008299, 100239862, 140669390, 1005499526, 10000442119, 10000761554, 10000853648, 10000973037, 10031199494, 10087799570, 1000006412206, 1090604591930, 1600005969190, 100000090745299, 100120849299260, 10000043099946481, 10078083499399210, 10442000392399960

Offset: 1

Klaus Brockhaus, Oct 20 2001

Integers like 196, for which a palindrome is supposedly never reached, are disregarded. A065199 gives the corresponding records.
a(39) <= N = 12000700000025339936491 for which A033665(N) = 288, found on April 26, 2019 according to Doucette's web site. - M. F. Hasler, Feb 16 2020
From A.H.M. Smeets, Sep 18 2021: (Start)
Let d_0 d_1 d_2 ... d_n be the decimal digits of an (n+1)-digit number.
All numbers in this sequence seem to satisfy the following condition:
d_0 = "1" or d_n = "9", and for all k, 0 < k < floor((n-1)/2), d_k = "0" or d_k = "9" or d_(n-k) = "0" or d_(n-k) = "9".
As from this, N = 12000700000025339936491, does not seem to be a record-setting number in this sequence, i.e., there must exist a smaller number N with at least a delay of 288 to reach a palindromic number. (End)

			Starting with 89, 24 'Reverse and Add' steps are needed to reach a palindrome; starting with n < 89, fewer (at most 6, in fact) steps are needed. So 89 is a term.

A.H.M. Smeets, Table of n, a(n) for n = 1..38 (terms n = 36..38 taken from Jason Doucette link below; offset adapted by _Georg Fischer_, Feb 17 2019)
Jason Doucette, World records
Ian J. Peter, Search for the biggest numeric palindrome, lost page, pointer to backup on web.archive.org as of July 2011.
Index entries for sequences related to Reverse and Add!

Cf. A033665, A033865, A023109, A065199.

limit = 10^3; (* Assumes that there is no palindrome if none is found before "limit" iterations *)
best = -1; Select[Range[0, 1000], (np = #; i = 0;
   While[np != IntegerReverse[np] && i < limit,
    np = np + IntegerReverse[np]; i++];
If[i >= limit, False, If[i > best, best = i; True]]) &] (* Robert Price, Oct 14 2019 *)

my(m, M=-1); for(n=0,oo, if(MA033665(n, M+39), print1(n","); M=m)) \\ Only for illustration, not suitable for producing terms > 10^6, even with the custom search limit given as optional 2nd arg to A033665. - M. F. Hasler, Feb 16 2020

Terms a(17) to a(21) from Sascha Kurz, Dec 05 2001
Terms a(22) ff. were taken from Jason Doucette, World records. - Klaus Brockhaus, Sep 24 2003
Offset changed to 1 by A.H.M. Smeets, Feb 14 2019
Edited by N. J. A. Sloane, Jul 16 2021

A065199 Record high values in A033665, ignoring those numbers that are believed never to reach a palindrome.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 24, 30, 53, 54, 55, 58, 64, 78, 79, 80, 82, 96, 97, 98, 109, 112, 113, 131, 135, 147, 149, 186, 187, 188, 198, 201, 232, 233, 236, 259, 260, 261

Offset: 1

Klaus Brockhaus, Oct 20 2001

Records for the number of 'Reverse and Add' steps needed to reach a palindrome.

A065198 gives the corresponding starting points.

			Starting with 89, 24 'Reverse and Add' steps are needed to reach a palindrome; starting with n < 89, at most 6 steps are needed.
For n = A065198(21) = 1005499526, a(21) = 109 "reverse and add" operations are needed to reach a palindrome; for all smaller n, at most 98 steps are needed.
For n = A065198(31) ~ 10^14, a(31) = 198 "reverse and add" operations are needed to reach a palindrome; for all smaller n, at most 188 steps are needed.
For n = A065198(36) ~ 10^18, a(36) = 259 "reverse and add" operations are needed to reach a palindrome; for all smaller n, at most 236 steps are needed.

Jason Doucette, World records
Index entries for sequences related to Reverse and Add!

Cf. A033665, A033865, A023109, A065198.

limit = 10^3; (* Assumes that there is no palindrome if none is found before "limit" iterations *)
best = -1; lst = {};
For[n = 0, n <= 100000, n++,
np = n; i = 0;
While[np != IntegerReverse[np] && i < limit,
  np = np + IntegerReverse[np]; i++];
If[i < limit && i > best, best = i; AppendTo[lst, i]]]; lst (* Robert Price, Oct 14 2019 *)

my(m, M=-1); for(n=0, oo, (MA033665(n, M+39))&&print1(M=m", ")) \\ For illustration; becomes very slow for terms > 70, even with the "custom" search limit as optional 2nd arg to A033665. - M. F. Hasler, Feb 16 2020

a(n) = A033665(A065198(n)). - M. F. Hasler, Feb 16 2020

Terms a(17) to a(21) from Sascha Kurz, Dec 05 2001
Terms a(22) onwards were taken from Jason Doucette, World records. - Klaus Brockhaus, Sep 24 2003
Terms a(36) to a(38) were taken from Jason Doucette, World records and added by A.H.M. Smeets, Feb 10 2019
Edited by N. J. A. Sloane, Jul 16 2021

A033665 Number of 'Reverse and Add' steps needed to reach a palindrome starting at n, or -1 if n never reaches a palindrome.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 0, 1, 2, 1, 2, 2, 1, 1, 1, 1, 1, 0, 1, 2, 2, 3, 1, 1, 1, 1, 2, 1, 0, 2, 3, 4, 1, 1, 1, 2, 1, 2, 2, 0, 4, 6, 1, 1, 2, 1, 2, 2, 3, 4, 0, 24, 1, 2, 1, 2, 2, 3, 4, 6, 24, 0, 1, 0, 1, 1

Offset: 0

N. J. A. Sloane

Palindromes themselves are not 'Reverse and Add!'ed, so they yield a zero!
Numbers n that may have a(n) = -1 (i.e., potential Lychrel numbers) appear in A023108. - Michael De Vlieger, Jan 11 2018
Record indices and values are given in A065198 and A065199. - M. F. Hasler, Feb 16 2020

			19 -> 19+91 = 110 -> 110+011 = 121 = palindrome, took 2 steps, so a(19)=2.
n = 89 needs 24 steps to end up with the palindrome 8813200023188. See A240510. - _Wolfdieter Lang_, Jan 12 2018

D. Wells, The Penguin Dictionary of Curious and Interesting Numbers Penguin Books, 1987, pp. 142-143.

Kerry Mitchell, Table of n, a(n) for n = 0..195
P. De Geest, Some thematic websources
Jason Doucette, World Records
S. K. Eddins, The Palindromic Order Of A Number [archived page]
Kerry Mitchell, Table of n, a(n) for n = 0..10000 (The -1 entries are only conjectural)
Kerry Mitchell, Table of n, a(n) for n = 0..100000 (The -1 entries are only conjectural)
I. Peter, Search for the biggest numeric palindrome [archived page]
T. Trotter, Jr., Palindrome Power [archived page]
Eric Weisstein's World of Mathematics, 196-Algorithm.
Index entries for sequences related to Reverse and Add!

Cf. A002113, A023108, A023109, A033865, A006960, A016016, A063048, A240510.
Equals A030547(n) - 1.
Cf. A065198, A065199 (record indices & values).

rev[n_]:=FromDigits[Reverse[IntegerDigits[n]]];radd[n_]:=n+rev[n];
pal[n_]:=If[n==rev[n],True,False];
raddN[n_]:=Length[NestWhileList[radd[#]&,n,pal[#]==False&]]-1;
raddN/@Range[0,195] (* Ivan N. Ianakiev, Aug 31 2015 *)
With[{nn = 10^3}, Array[-1 + Length@ NestWhileList[# + IntegerReverse@ # &, #, !PalindromeQ@ # &, 1, nn] /. k_ /; k == nn -> -1 &, 200]] (* Michael De Vlieger, Jan 11 2018 *)

rev(n)={d=digits(n);p="";for(i=1,#d,p=concat(Str(d[i]),p));return(eval(p))}
a(n)=if(n==rev(n),return(0));for(k=1,10^3,i=n+rev(n);if(rev(i)==i,return(k));n=i)
n=0;while(n<100,print1(a(n),", ");n++) \\ Derek Orr, Jul 28 2014

A033665(n,LIM=333)={-!for(i=0,LIM,my(r=A004086(n)); n==r&&return(i); n+=r)} \\ with {A004086(n)=fromdigits(Vecrev(digits(n)))}. The second optional arg is a search limit that could be taken smaller up to very large n, e.g., 99 for n < 10^9, 200 for n < 10^14, 250 for n < 10^18: see A065199 for the records and A065198 for the n's. - M. F. Hasler, Apr 13 2019, edited Feb 16 2020

A033665 = lambda n, LIM=333: next((i for i in range(LIM) if is_A002113(n) or not(n := A004086(n)+n)), -1) # The second, optional argument is a search limit, see above. - M. F. Hasler, May 23 2024

More terms from Patrick De Geest, Jun 15 1998

I truncated the b-file at n=195, since the value of a(196) is not presently known (cf. A006960). The old b-files are now a-files. - N. J. A. Sloane, May 09 2015

A016016 Number of iterations of Reverse and Add which lead to a palindrome, or -1 if no palindrome is ever reached.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 1, 1, 1, 1, 2, 1, 2, 2, 3, 4, 1, 1, 1, 2, 1, 2, 2, 3, 4, 6, 1, 1, 2, 1, 2, 2, 3, 4, 6, 24, 1, 2, 1, 2, 2, 3, 4, 6, 24

Offset: 1

Robert G. Wilson v

A first 'Reverse and Add' operation is always made, even if the starting value n is already a palindrome, in contrast to the variant A033665.
It is conjectured that a(196) = -1, see A023108.
Because A061563 has offset 0 one should add a(0) = 1 here. - Wolfdieter Lang, Jan 12 2018
Record indices and values beyond a(1) = 1 and a(5) = 2 are given in A065198 and A065199: These refer to the variant A033665 (main entry with more up-to-date references), as can be seen from A065199(1..3) = (0, 1, 2) for A065198(1..3) = (0, 10, 19). But all larger records correspond to a non-palindromic index n, in which case a(n) = A033665(n). - M. F. Hasler, Feb 16 2020

			6 -> 6 + 6 = 12 -> 12 + 21 = 33 is palindromic, took 2 steps so a(6)=2.
n = 89 needs 24 steps to end up with the palindrome 8813200023188. See A240510. - _Wolfdieter Lang_, Jan 12 2018

T. D. Noe, Table of n, a(n) for n = 1..195
J. Walker, Three Years Of Computing: Final Report On The Palindrome Quest
Eric Weisstein's World of Mathematics, 196-Algorithm.
Index entries for sequences related to Reverse and Add!

Cf. A033665, A023109, A006960, A061563, A240510.

tol = 1000; r[n_] := FromDigits[Reverse[IntegerDigits[n]]]; palQ[n_] := n == r[n]; ar[n_] := n + r[n]; Table[k = 0; If[palQ[n], n = ar[n]; k = 1]; While[! palQ[n] && k < tol, n = ar[n]; k++]; If[k == tol, k = -1]; k, {n, 98}] (* Jayanta Basu, Jul 11 2013 *)
With[{nn = 10^3}, Array[-1 + Length@ NestWhileList[# + IntegerReverse@ # &, #, ! PalindromeQ@ # &, {2, 1}, 10^3] /. k_ /; k == nn -> -1 &, 200, 0]] (* Michael De Vlieger, Jan 11 2018 *)

a(n) = my(x=n, i=0); while(1, x=x+eval(concat(Vecrev(Str(x)))); i++; if(x==eval(concat(Vecrev(Str(x)))), return(i))) \\ Felix Fröhlich, Jan 12 2018

A016016(n, LIM=exponent(n+1)*5)={-!for(i=0, LIM, my(r=A004086(n)); n==r&&i&&return(i); n+=r)} \\ with {A004086(n)=fromdigits(Vecrev(digits(n)))}. The second optional arg is a search limit, with default value chosen according to known records A065199 and indices A065198. - M. F. Hasler, Feb 16 2020

A061563 Start with n; add to itself with digits reversed; if palindrome, stop; otherwise repeat; a(n) gives palindrome at which it stops, or -1 if no palindrome is ever reached.

Original entry on oeis.org

0, 2, 4, 6, 8, 11, 33, 55, 77, 99, 11, 22, 33, 44, 55, 66, 77, 88, 99, 121, 22, 33, 44, 55, 66, 77, 88, 99, 121, 121, 33, 44, 55, 66, 77, 88, 99, 121, 121, 363, 44, 55, 66, 77, 88, 99, 121, 121, 363, 484, 55, 66, 77, 88, 99, 121, 121, 363, 484, 1111, 66, 77, 88, 99, 121

Offset: 0

N. J. A. Sloane, May 18 2001

It is believed that n = 196 is the smallest integer which never reaches a palindrome.

			19 -> 19 + 91 = 110 -> 110 + 011 = 121, so a(19) = 121.

Cf. A033865. A016016 (number of steps), A023109, A006950, A023108.

var st: stack; test: boolean; end; for k := 0 to 60 do n := k; test := true; while test do n := n + int_reverse(n); test := n <> int_reverse(n); end; stack_push(st,n); end; stack2array(st);

tol = 1000; r[n_] := FromDigits[Reverse[IntegerDigits[n]]]; palQ[n_] := n == r[n]; ar[n_] := n + r[n]; Table[k = 0; If[palQ[n], n = ar[n]; k = 1]; While[! palQ[n] && k < tol, n = ar[n]; k++]; If[k == tol, n = -1]; n, {n, 0, 64}] (* Jayanta Basu, Jul 11 2013 *)
Table[Module[{k=n+IntegerReverse[n]},While[k!=IntegerReverse[k],k=k+IntegerReverse[k]];k],{n,0,70}] (* The program uses the IntegerReverse function from Mathematica version 10 *) (* Harvey P. Dale, Jul 19 2016 *)

Corrected and extended by Klaus Brockhaus, May 20 2001

More terms from Ray Chandler, Jul 25 2003

A281301 Trajectory of 1000004999700144385 under the "Reverse and Add!" operation.

Original entry on oeis.org

1000004999700144385, 6834415079694144386, 13668830049399288772, 41457129443403175403, 71914259877895350817, 143719619755790592734, 581014717313707510075, 1151030424627424920260, 1771324671891665221771, 3542550333873429453542, 5996099577656760005995

Offset: 0

Andrey S. Shchebetov and Sergei D. Shchebetov, Jan 21 2017

1000004999700144385 is the largest of the first 225 numbers that require exactly 259 steps to turn into a palindrome (see A281390). The sequence reaches a 119-digit palindrome after 259 steps (see b-file). The number was obtained empirically using computer algorithms and was not reported before.

Row 1000004999700144385 of the array in A243238. - Felix Fröhlich, Jan 21 2017

			a(1) = 1000004999700144385 + 5834410079994000001 = 6834415079694144386.

Popular Computing (Calabasas, CA), The 196 Problem, Vol. 3 (No. 30, Sep 1975).

Sergei D. Shchebetov, Table of n, a(n) for n = 0..259
Jason Doucette, World Records
Yutaka Nishiyama, Numerical Palindromes and the 196 Problem, International Journal of Pure and Applied Mathematics, Volume 80, No. 3, 2012, 375-384.
R. Styer, The Palindromic Conjecture and the Fibonacci Sequence, Villanova University, 1986, 1-11.
C. W. Trigg, Palindromes by Addition, Mathematics Magazine, 40 (1967), 26-28.
C. W. Trigg, More on Palindromes by Reversal-Addition, Mathematics Magazine, 45 (1972), 184-186.
Wikipedia, Lychrel Number
196 and Other Lychrel Numbers, 196 and Lychrel Number
Index entries for sequences related to Reverse and Add!

Cf. A023109, A033672, A065198, A065199, A065320, A065321, A065322, A065323, A065324, A065325, A065326, A065327, A070743, A072216, A072217, A072218, A243238, A281390.

NestList[#+IntegerReverse[#]&,1000004999700144385,10] (* Harvey P. Dale, Dec 24 2021 *)

terms(n) = my(x=1000004999700144385, i=0); while(1, print1(x, ", "); x=x+eval(concat(Vecrev(Str(x)))); i++; if(i==n, break))
/* Print initial 9 terms as follows: */
terms(9) \\ Felix Fröhlich, Jan 21 2017

a(n+1) = a(n) + rev(a(n)).

a(9)-a(10) from Felix Fröhlich, Jan 21 2017

A281390 Numbers which require exactly 259 'Reverse and Add' steps to reach a palindrome.

Original entry on oeis.org

1000000079994144385, 1000000089894144385, 1000000099794144385, 1000000179984144385, 1000000189884144385, 1000000199784144385, 1000000279974144385, 1000000289874144385, 1000000299774144385, 1000000379964144385, 1000000389864144385, 1000000399764144385

Offset: 1

Andrey S. Shchebetov and Sergei D. Shchebetov, Jan 21 2017

The sequence starts with 1000000079994144385 (the 19-digit number discovered by Vaughn Suite on Jul 26 2005 and rediscovered by Jason Doucette on Nov 28 2005) and continues for another 224 terms (none previously reported) each turning into a 119-digit palindrome after 259 steps until the sequence ends with 1000004999700144385. The distance between successive terms in the reported sequence has 9000000 as the greatest common divisor. No further numbers beyond 1000004999700144385 belonging to the same sequence are known, discovered or reported. The sequence was found empirically using computer modeling algorithms.

The sequence was extended to 1620000 terms in total and currently ends with 6834414999700000000 (see a-file). The sequence is complete - no further numbers beyond 6834414999700000000 belonging to the same sequence exist. The sequence was predicted theoretically and found empirically using computer modeling algorithms. - Sergei D. Shchebetov, May 12 2017

			Each term requires exactly 259 steps to turn into a 119-digit palindrome, the last term of A281301, and is separated by some multiples of 9000000 from the adjacent sequence terms.

Popular Computing (Calabasas, CA), The 196 Problem, Vol. 3 (No. 30, Sep 1975).

Sergei D. Shchebetov, Table of n, a(n) for n = 1..225
Jason Doucette, World Records
Yutaka Nishiyama, Numerical Palindromes and the 196 Problem, International Journal of Pure and Applied Mathematics, Volume 80 No. 3 2012, 375-384.
Sergei D. Shchebetov, 1620000 terms (zipped file)
R. Styer, The Palindromic Conjecture and the Fibonacci Sequence, Villanova University, 1986, 1-11.
C. W. Trigg, Palindromes by Addition, Mathematics Magazine, 40 (1967), 26-28.
C. W. Trigg, More on Palindromes by Reversal-Addition, Mathematics Magazine, 45 (1972), 184-186.
Wikipedia, Lychrel Number
196 and Other Lychrel Numbers, 196 and Lychrel Number
Index entries for sequences related to Reverse and Add!

Cf. A023109, A033672, A065198, A065199, A065320, A065321, A065322, A065323, A065324, A065325, A065326, A065327, A070743, A072216, A072217, A072218, A281301.

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).

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A006960 Reverse and Add! sequence starting with 196.

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A033865 Start with n; if palindrome, stop; otherwise add to itself with digits reversed; a(n) gives palindrome at which it stops, or -1 if no palindrome is ever reached.

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A065198 Indices of record high values in A033665, ignoring those numbers that are believed never to reach a palindrome.

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A065199 Record high values in A033665, ignoring those numbers that are believed never to reach a palindrome.

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A033665 Number of 'Reverse and Add' steps needed to reach a palindrome starting at n, or -1 if n never reaches a palindrome.

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