A033665 -id:A033665 - cp's OEIS frontend

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

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

A088753 Numbers k such that the Reverse and Add! trajectory of k (presumably) does not reach a palindrome (with the exception of k itself) and does not join the trajectory of any term m < k.

Original entry on oeis.org

196, 879, 1997, 7059, 9999, 10553, 10563, 10577, 10583, 10585, 10638, 10663, 10668, 10697, 10715, 10728, 10735, 10746, 10748, 10783, 10785, 10787, 10788, 10877, 10883, 10963, 10965, 10969, 10977, 10983, 10985, 12797, 12898, 13097, 13197, 13694, 14096, 14698, 15297, 15597, 18598, 18798

Offset: 1

Klaus Brockhaus, Nov 04 2003

Although the starting number k is regarded as part of the trajectory, it is allowed to be palindromic. Hence palindromes are not excluded from the sequence. A063048 is obtained if palindromes are excluded. The smallest term in A088753 but not in A063048 is 9999, the smallest term in A063048 but not in A088753 is 19098.
W. VanLandingham and others have computed nearly 10^7 terms (all terms < 10^14), cf. W. VanLandingham, 196 and Other Lychrel Numbers.
From M. F. Hasler, Apr 13 2019: (Start)
Lychrel numbers listed here are also called "seeds", in contrast to Kin numbers A023108 which include all terms in the orbits of the former.
It is not easy to determine whether the orbit of a given term will never merge into the orbit of an earlier term. It seems that the property of "disjoint orbit" is as conjectural as the property of not reaching a palindrome. One could specify a "search limit" in order to get a well-defined sequence. The given list of terms has been checked and extended by considering the orbits up to members of size <= 10^199 at least. Given that the number increases by a factor 10 roughly every 2.416 iterations, this corresponds to about 500 iterations. (End)

			From _M. F. Hasler_, Apr 13 2019: (Start)
All numbers < 196 quickly reach a palindrome under iterations of the reverse-and-add function A056964, cf. A033665.
a(1) = 196 is the smallest integer which appears to never reach a palindrome (checked up to 10^9 iterations!).
Next, A056964(196) = 196 + 691 = 887 is in the orbit of 196 and will therefore never reach a palindrome if 196 does not. However, we do not list this term in this sequence because it is in the orbit of the smaller term 196.
Similarly, 295 + 592 = 887 = A056964(196). Therefore, 295 will also never reach a palindrome if 196 (and therefore 887) doesn't. But again we will not list this number, because its orbit merges into that of the smaller term 196.
The next number which appears to be a Lychrel and has an orbit (conjectured to be) disjoint with that of 196 is 897 = a(2). (End)

M. F. Hasler, Table of n, a(n) for n = 1..74 (all terms up to 10^5), Apr 13 2019.
W. VanLandingham, 196 and Other Lychrel Numbers [Copy on web.archive.org as of 05/2018, original site p196.org seems no longer available. - _M. F. Hasler_, Apr 13 2019]

Cf. A063048 (variant excluding palindromes), A023108 (Kin numbers), A056964 (reverse-and-add), A006960 (orbit of 196), A033665 (steps to reach a palindrome), A061563 (terminating palindrome of n's orbit), A002113 (palindromes).

limit = 10^3; (* Assumes that there is no palindrome if none is found before "limit" iterations *)
utraj = {};
Select[Range[0,
  20000], (np = # + IntegerReverse[#];
   x = NestWhileList[ # + IntegerReverse[#] &, np, ! PalindromeQ[#] &, 1, limit];
   If[Length[x] >= limit  && Intersection[x, utraj] == {},
    utraj = Union[utraj, x, {np}]; True,
utraj = Union[utraj, x, {np}]]) &] (* Robert Price, Oct 16 2019 *)

A088753_upto(LIM=2e4,M=1e199)={my(U=[],a=List());for(n=1,LIM, my(t=n); while( tA002113(t=A056964(t)) && next(2)); setsearch(U,t) && next; U=setunion(U,[t]); print1(n","); listput(a,n)); Set(a)} \\ M. F. Hasler, Apr 13 2019

Edited by M. F. Hasler, Apr 13 2019

A033670 Trajectory of 89 under map x->x + (x-with-digits-reversed).

Original entry on oeis.org

89, 187, 968, 1837, 9218, 17347, 91718, 173437, 907808, 1716517, 8872688, 17735476, 85189247, 159487405, 664272356, 1317544822, 3602001953, 7193004016, 13297007933, 47267087164, 93445163438

Offset: 0

N. J. A. Sloane

The sequence reaches a palindrome at a(24)= 8813200023188 (cf. A033665). - Klaus Brockhaus, Jun 07 2002

			a(3) = 968 because 187 + 781 = 968

Popular Computing (Calabasas, CA), The 196 Problem, Vol. 3 (No. 30, Sep 1975), page PC30-6. Gives full 25-term trajectory of 89.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jason Doucette, World Records
I. Peter, More trajectories
Index entries for sequences related to Reverse and Add!

NestList[# + FromDigits[Reverse[IntegerDigits[#]]]&, 89, 40] (* Vincenzo Librandi, May 03 2014 *)

A065207 Numbers which need two 'Reverse and Add' steps to reach a palindrome.

Original entry on oeis.org

19, 28, 37, 39, 46, 48, 49, 57, 58, 64, 67, 73, 75, 76, 82, 84, 85, 91, 93, 94, 109, 119, 129, 139, 149, 150, 152, 153, 154, 159, 160, 162, 163, 169, 170, 172, 173, 179, 189, 208, 218, 219, 228, 229, 238, 239, 248, 250, 251, 253, 258, 259, 260, 261, 268, 269

Offset: 1

Klaus Brockhaus, Oct 21 2001

The number of steps starts at 0, so palindromes (cf. A002113) are excluded.

Numbers k such that A033665(k) = 2. - Andrew Howroyd, Dec 06 2024

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)
Index entries for sequences related to Reverse and Add!

Cf. A002113, A015977, A033665, A065206.

revadd_steps(2,58); (* For the definition of function revadd_steps see A065206. *)

trasQ[n_]:=Length[NestWhileList[IntegerReverse[#]+#&,n,!PalindromeQ[ #]&,1,5]] ==3; Select[Range[300],trasQ] (* Harvey P. Dale, Apr 13 2022 *)

isok(n,s=2)={for(k=0, s, my(r=fromdigits(Vecrev(digits(n)))); if(r==n, return(k==s)); n += r); 0} \\ Andrew Howroyd, Dec 06 2024

def ra(n): s = str(n); return int(s) + int(s[::-1])
def ispal(n): s = str(n); return s == s[::-1]
def aupto(limit):
  alst = []
  for k in range(limit+1):
    if ispal(k): continue
    k2 = ra(k)
    if ispal(k2): continue
    if ispal(ra(k2)): alst.append(k)
  return alst
print(aupto(269)) # Michael S. Branicky, May 06 2021

Offset changed from 0 to 1 by Harry J. Smith, Oct 14 2009

A193239 Number of "Reverse and Add" steps needed to reach a palindrome using the complex base -1+i, or -1 if a palindrome is never reached.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, -1, 0, 1, 0, 1, 6, 1, 6, 1, 0, 1, 0, 1, 4, -1, 0, -1, -1, 1, 4, -1, 0, -1, -1, 5, 0, 1, 0, 1, 6, 1, -1, 1, -1, 1, -1, 1, -1, -1, 0, 7, -1, 1, 6, 7, 0, 1, -1, 1, 2, 1, -1, 1, 2, 7, -1, -1, 0, 1, 0, 1, -1, 1, -1, 1, -1, 3, 0, -1, -1, 9, 2, 1

Offset: 0

Kerry Mitchell, Jul 19 2011

N is converted to its binary representation before iterating.

			Decimal 2 is 10 in binary, which is -1+i using complex base -1+i. Reversing 10 gives 01, or 1+0i.  Adding both results in 0+i, or 11 using the complex base, which is a palindrome.  Decimal 2 took 1 step to reach a palindrome, so a(2) = 1.

Kerry Mitchell, Table of n, a(n) for n = 0..10000
W. J. Gilbert, Arithmetic in Complex Bases, Mathematics Magazine, Vol. 57, No. 2 (Mar., 1984), pp. 77-81.

Cf. A033665 gives the steps to reach a palindrome in base 10.

A065208 Numbers which need three 'Reverse and Add' steps to reach a palindrome.

Original entry on oeis.org

59, 68, 86, 95, 155, 156, 157, 158, 164, 165, 168, 178, 180, 184, 185, 186, 194, 199, 249, 254, 255, 256, 257, 263, 264, 267, 277, 283, 284, 285, 293, 298, 299, 348, 349, 354, 355, 356, 362, 366, 376, 382, 384, 389, 392, 397, 398, 399, 439, 447, 448, 449, 452, 453, 455, 461, 462, 465, 475, 481, 482

Offset: 1

Klaus Brockhaus, Oct 21 2001

The number of steps starts at 0, so palindromes (cf. A002113) are excluded.

Numbers k such that A033665(k) = 3. - Andrew Howroyd, Dec 06 2024

Harry J. Smith, Table of n, a(n) for n = 1..1000
Index entries for sequences related to Reverse and Add!

Cf. A002113, A015979, A033665, A065206.

revadd_steps(3,54). For the definition of function revadd_steps see A065206.

palQ[n_]:=Module[{idn=IntegerDigits[n]},idn==Reverse[idn]]
tst[n_]:=palQ/@NestList[#+FromDigits[Reverse[IntegerDigits[#]]]&,n,3]=={False,False,False,True}
Select[Range[750],tst] (* Harvey P. Dale, Nov 26 2010 *)

isok(n,s=3)={for(k=0, s, my(r=fromdigits(Vecrev(digits(n)))); if(r==n, return(k==s)); n += r); 0} \\ Andrew Howroyd, Dec 06 2024

Offset changed from 0 to by Harry J. Smith, Oct 14 2009

More terms from Harvey P. Dale, Nov 26 2010

A065209 Numbers which need four 'Reverse and Add' steps to reach a palindrome.

Original entry on oeis.org

69, 78, 87, 96, 174, 175, 183, 192, 195, 273, 274, 280, 290, 291, 294, 372, 381, 390, 471, 472, 480, 492, 539, 570, 571, 579, 591, 599, 629, 638, 649, 670, 678, 679, 690, 698, 699, 728, 729, 748, 749, 769, 778, 789, 798, 819, 827, 836, 839, 847, 876, 877

Offset: 1

Klaus Brockhaus, Oct 21 2001

The number of steps starts at 0, so palindromes (cf. A002113) are excluded.

Numbers k such that A033665(k) = 4. - Andrew Howroyd, Dec 06 2024

Andrew Howroyd, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Reverse and Add!

Cf. A002113, A015980, A033665, A065206.

select(k->isok(k,4), [1..900]) \\ isok defined in A065206. - Andrew Howroyd, Dec 06 2024

Offset changed from 0 to 1 by Harry J. Smith, Oct 14 2009

A065210 Numbers which need five 'Reverse and Add' steps to reach a palindrome.

Original entry on oeis.org

166, 176, 198, 265, 275, 297, 364, 374, 396, 463, 473, 495, 549, 562, 572, 594, 639, 648, 661, 671, 693, 738, 760, 770, 792, 837, 846, 891, 936, 945, 990, 1396, 1486, 1576, 1666, 1697, 1756, 1787, 1791, 1793, 1796, 1846, 1877, 1883, 1886, 1890, 1936

Offset: 1

Klaus Brockhaus, Oct 21 2001

The number of steps starts at 0, so palindromes (cf. A002113) are excluded.

Numbers k such that A033665(k) = 5. - Andrew Howroyd, Dec 06 2024

Harvey P. Dale, Table of n, a(n) for n = 1..3000
Index entries for sequences related to Reverse and Add!

Cf. A002113, A015982, A033665, A065206.

palQ[k_]:=IntegerDigits[k]==Reverse[IntegerDigits[k]]; fraQ[n_]:= Module[ {ras=NestList[#+FromDigits[Reverse[IntegerDigits[#]]]&,n,5]},palQ/@ ras=={False,False,False,False,False,True}]; Select[Range[2000],fraQ] (* Harvey P. Dale, Sep 28 2015 *)

Offset changed to 1 by Harry J. Smith, Oct 14 2009

A065211 Numbers which need six 'Reverse and Add' steps to reach a palindrome.

Original entry on oeis.org

79, 97, 182, 281, 380, 779, 799, 889, 977, 988, 997, 1069, 1079, 1159, 1169, 1249, 1259, 1339, 1349, 1429, 1439, 1519, 1529, 1609, 1619, 1699, 1709, 1789, 1799, 1879, 1889, 1896, 1969, 1979, 1986, 2059, 2068, 2078, 2089, 2149, 2158, 2168, 2179, 2239

Offset: 1

Klaus Brockhaus, Oct 21 2001

The number of steps starts at 0, so palindromes (cf. A002113) are excluded.

Numbers k such that A033665(k) = 6. - Andrew Howroyd, Dec 06 2024

Andrew Howroyd, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Reverse and Add!

Cf. A002113, A015984, A033665, A065206.

palQ[n_] := Module[{idn = IntegerDigits[n]}, idn == Reverse[idn]]; pal6Q[ n_]:= Module[{c=NestList[#+FromDigits[Reverse[IntegerDigits[#]]]&, n,6]}, palQ/@c=={False,False,False,False,False,False,True}]; Select[Range[ 2300],pal6Q] (* Harvey P. Dale, Sep 09 2012 *)

Offset changed from 0 to 1 by Harry J. Smith, Oct 14 2009

A065212 Numbers which need seven 'Reverse and Add' steps to reach a palindrome.

Original entry on oeis.org

188, 190, 197, 287, 296, 386, 395, 485, 584, 593, 683, 692, 782, 791, 881, 890, 980, 1394, 1484, 1574, 1664, 1754, 1844, 1898, 1934, 1988, 1992, 1994, 1999, 2393, 2483, 2573, 2663, 2753, 2843, 2897, 2933, 2987, 2991, 2993, 2998, 3089, 3179, 3269, 3359

Offset: 1

Klaus Brockhaus, Oct 21 2001

The number of steps starts at 0, so palindromes (cf. A002113) are excluded.

Numbers k such that A033665(k) = 7. - Andrew Howroyd, Dec 08 2024

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for sequences related to Reverse and Add!

Cf. A002113, A015986, A033665, A065206.

lenQ[n_]:= Length[NestWhileList[# + FromDigits[Reverse[IntegerDigits[#]]]&, n,#!= FromDigits[Reverse[IntegerDigits[#]]]&, 1, 10]] == 8; Select[Range[500], lenQ] (* Vincenzo Librandi, Sep 24 2013 *)

Offset changed from 0 to 1 by Harry J. Smith, Oct 14 2009

cp's OEIS Frontend

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

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A088753 Numbers k such that the Reverse and Add! trajectory of k (presumably) does not reach a palindrome (with the exception of k itself) and does not join the trajectory of any term m < k.

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A033670 Trajectory of 89 under map x->x + (x-with-digits-reversed).

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A065207 Numbers which need two 'Reverse and Add' steps to reach a palindrome.

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A193239 Number of "Reverse and Add" steps needed to reach a palindrome using the complex base -1+i, or -1 if a palindrome is never reached.

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A065208 Numbers which need three 'Reverse and Add' steps to reach a palindrome.

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A065209 Numbers which need four 'Reverse and Add' steps to reach a palindrome.

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A065210 Numbers which need five 'Reverse and Add' steps to reach a palindrome.

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