A006960 -id:A006960 - cp's OEIS frontend

A072217 Consider the Reverse and Add! problem (cf. A001127); of all the n-digit numbers N which eventually reach a palindrome, pick that number N which takes the greatest number of steps to converge (in case of a tie, pick the smallest N); sequence gives number of steps N takes to converge.

2, 24, 23, 21, 55, 64, 96, 96, 98, 109, 149, 149, 188, 186, 201, 197, 236, 232

Offset: 1

N. J. A. Sloane, Jul 05 2002

Since we do not even know if 196 eventually converges (see A006960, A023108) for n >= 3 these values are only conjectures.

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

Cf. A072216, A072218, A001127, A006960, A023108.

Corrected and extended by Jason Doucette, Mar 29 2005; Oct 09 2005

A072218 Consider the Reverse and Add! problem (cf. A001127); of all the n-digit numbers N which eventually reach a palindrome, pick that number N which takes the greatest number of steps to converge (in case of a tie, pick the smallest N); sequence gives palindrome that is reached.

Original entry on oeis.org

11, 8813200023188, 8813200023188, 8813200023188, 4668731596684224866951378664, 682049569465550121055564965940286, 555458774083726674580862268085476627380477854555, 555458774083726674580862268085476627380477854555, 1345428953367763125675365555635765213677633598245431

Offset: 1

N. J. A. Sloane, Jul 05 2002

Since we do not even know if 196 eventually converges (see A006960, A023108) for n >= 3 these values are only conjectures.

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

Cf. A072216, A072217, A001127, A006960, A023108.

A243238 Table T(n,r) of terms in the reverse and add sequences of positive integers n read by antidiagonals.

Original entry on oeis.org

1, 2, 2, 4, 4, 3, 8, 8, 6, 4, 16, 16, 12, 8, 5, 77, 77, 33, 16, 10, 6, 154, 154, 66, 77, 11, 12, 7, 605, 605, 132, 154, 22, 33, 14, 8, 1111, 1111, 363, 605, 44, 66, 55, 16, 9, 2222, 2222, 726, 1111, 88, 132, 110, 77, 18, 10, 4444, 4444, 1353, 2222, 176, 363, 121, 154, 99, 11, 11

Offset: 1

Felix Fröhlich, Jun 12 2014

			T(5,6) = 88, since 88 is the 6th term in the reverse and add sequence of 5.
Table starts with:
  1 2 4 8 16 77 154 605 1111 2222
  2 4 8 16 77 154 605 1111 2222 4444
  3 6 12 33 66 132 363 726 1353 4884
  4 8 16 77 154 605 1111 2222 4444 8888
  5 10 11 22 44 88 176 847 1595 7546
  6 12 33 66 132 363 726 1353 4884 9768
  7 14 55 110 121 242 484 968 1837 9218
  8 16 77 154 605 1111 2222 4444 8888 17776
  9 18 99 198 1089 10890 20691 40293 79497 158994
  10 11 22 44 88 176 847 1595 7546 14003

Alois P. Heinz, Antidiagonals n = 1..141, flattened
Index entries for sequences related to Reverse and Add!

Rows n=1, 3, 5, 7, 9 give: A001127, A033648, A033649, A033650, A033651.
Cf. A006960, A023108, A033670, A088753, A089694, A240510.
Main diagonal gives A244058.

T:= proc(n, r) option remember; `if`(r=1, n, (h-> h +(s->
      parse(cat(s[-i]$i=1..length(s))))(""||h))(T(n, r-1)))
    end:
seq(seq(T(n, 1+d-n), n=1..d), d=1..12);  # Alois P. Heinz, Jun 18 2014

rad[n_] := n + FromDigits[Reverse[IntegerDigits[n]]];
T[n_, 1] := n; T[n_, k_] := T[n, k] = rad[T[n, k-1]];
Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Apr 08 2016 *)

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

A033650 Trajectory of 7 under map x --> x + (x-with-digits-reversed).

Original entry on oeis.org

7, 14, 55, 110, 121, 242, 484, 968, 1837, 9218, 17347, 91718, 173437, 907808, 1716517, 8872688, 17735476, 85189247, 159487405, 664272356, 1317544822, 3602001953, 7193004016, 13297007933, 47267087164, 93445163438, 176881317877, 955594506548, 1801200002107

Offset: 0

N. J. A. Sloane

A Reverse and Add! sequence.
Trajectories of 19, 23, 28, 29, 32, 37, 38, 41, 46, 47, 49, 50, ..., merge into this sequence. - Robert G. Wilson v, Dec 16 2005
A Reverse and Add! sequence.

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 251 terms from Reinhard Zumkeller)
Index entries for sequences related to Reverse and Add!

Cf. A006960, A001127, A033648, A033649, A033651.
Cf. A056964, A004086.
Row n=7 of A243238.

a033650 n = a033650_list !! n
a033650_list = iterate a056964 7 -- Reinhard Zumkeller, Sep 22 2011

a:= proc(n) option remember; `if`(n=0, 7, (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, Jun 18 2014

NestList[ # + FromDigits@Reverse@IntegerDigits@# &, 7, 26] (* Robert G. Wilson v *)

A033651 Trajectory of 9 under map x->x + (x-with-digits-reversed).

Original entry on oeis.org

9, 18, 99, 198, 1089, 10890, 20691, 40293, 79497, 158994, 658845, 1207701, 2284722, 4559544, 9019098, 17928207, 88211178, 175322466, 839546037, 1570191975, 7362102726, 13634115363, 49985258994, 99970517988, 188942025987, 978462275868, 1847034540747, 9317488848228

Offset: 0

N. J. A. Sloane

Trajectories of 27, 36, 45, 54, 63, 72, 81, 90, ..., merge into this sequence. - Robert G. Wilson v, Dec 16 2005.

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 251 terms from Reinhard Zumkeller)
Index entries for sequences related to Reverse and Add!

Cf. A006960, A001127, A033648, A033649, A033650.
Cf. A056964, A004086.
Row n=9 of A243238.
Partial sums of A112296.

a063051 n = a063051_list !! n
a063051_list = iterate a056964 879 -- Reinhard Zumkeller, Sep 22 2011

a:= proc(n) option remember; `if`(n=0, 9, (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, Jun 18 2014

NestList[ # + FromDigits@Reverse@IntegerDigits@# &, 9, 26] (* Robert G. Wilson v *)

A033649 Trajectory of 5 under map x->x + (x-with-digits-reversed).

Original entry on oeis.org

5, 10, 11, 22, 44, 88, 176, 847, 1595, 7546, 14003, 44044, 88088, 176176, 847847, 1596595, 7553546, 14007103, 44177144, 88354288, 176599676, 853595347, 1597190705, 6668108656, 13236127322, 35608290553

Offset: 0

N. J. A. Sloane

Trajectories of 15, 21, 24, 30, 39, 42, 48, 51, 57, 60, 69, 75, 78, 84, 87, 93, 96, ..., merge into this sequence. - Robert G. Wilson v, Dec 16 2005
A Reverse and Add! sequence.
Trajectories of 13, 17, 20, 26, 31, 35, 40, 53, 62, 71, 79, 80, 97, ..., merge into this sequence. - Robert G. Wilson v, Dec 16 2005

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 251 terms from Reinhard Zumkeller)
Index entries for sequences related to Reverse and Add!

Cf. A006960, A001127, A033648, A033650, A033651.
Cf. A056964, A004086.
Row n=5 of A243238.

a033649 n = a033649_list !! n
a033649_list = iterate a056964 5 -- Reinhard Zumkeller, Sep 22 2011

a:= proc(n) option remember; `if`(n=0, 5, (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, Jun 18 2014

NestList[ # + FromDigits@Reverse@IntegerDigits@# &, 5, 29] (* Robert G. Wilson v, Dec 16 2005 *)

A063433 'Reverse and Add!' trajectory of 10577.

Original entry on oeis.org

10577, 88078, 175166, 836737, 1574375, 7309126, 13528163, 49710694, 99312488, 187733887, 976071668, 1842242347, 9274664828, 17559329557, 93151725128, 175304440267, 937348843838, 1775697687577, 9533565653348, 17967131306707

Offset: 0

Klaus Brockhaus, Jul 20 2001

			a(1) = 10577 + 77501 = 88078.

Harry J. Smith, Table of n, a(n) for n=0,...,200
Index entries for sequences related to Reverse and Add!

A033865, A023108, A006960, A063434.

Cf. A056964, A004086.

m := 10577; stop := 20; c := 0; rev := int_reverse(m); while m <> rev and c < stop do inc(c); write(m," "); m := m + rev; rev := int_reverse(m); end;

a063433 n = a063433_list !! n
a063433_list = iterate a056964 10577 -- Reinhard Zumkeller, Sep 22 2011

NestList[#+FromDigits[Reverse[IntegerDigits[#]]]&,10577, 20]  (* Harvey P. Dale, Apr 03 2011 *)

Rev(x)= { local(d,r); r=0; while (x>0, d=x-10*(x\10); x\=10; r=r*10 + d); return(r) }
{ for (n=0, 200, if (n, a+=Rev(a), a=10577); write("b063433.txt", n, " ", a) ) } \\ Harry J. Smith, Aug 21 2009

A063051 'Reverse and Add!' trajectory of 879.

Original entry on oeis.org

879, 1857, 9438, 17787, 96558, 182127, 903408, 1707717, 8884788, 17759676, 85455447, 159910905, 668930856, 1326970722, 3597766953, 7194444906, 13288889823, 46187778054, 91275556218, 172541113437, 906852258708

Offset: 0

Klaus Brockhaus, Jul 07 2001

			a(1) = 879 + 978 = 1857.

Harry J. Smith and Michael Lee, Table of n, a(n) for n = 0..2366 (Harry J. Smith provided terms 0 through 200)
Index entries for sequences related to Reverse and Add!

Cf. A033865, A023108, A006960, A063052, A056964, A004086.

m := 879; stop := 25; c := 0; rev := int_reverse(m); while m <> rev and c < stop do inc(c); write(m," "); m := m + rev; rev := int_reverse(m); end;

a033651 n = a033651_list !! n
a033651_list = iterate a056964 9 -- Reinhard Zumkeller, Sep 22 2011

NestList[# + FromDigits[Reverse[IntegerDigits[#]]]&, 879, 40] (* Vincenzo Librandi, Sep 23 2013 *)

Rev(x)= { local(d); r=0; while (x>0, d=x-10*(x\10); x\=10; r=r*10 + d); return(r) }
{ for (n=0, 200, if (n, a+=Rev(a), a=879); write("b063051.txt", n, " ", a) ) } \\ Harry J. Smith, Aug 16 2009

A063057 'Reverse and Add!' trajectory of 7059.

Original entry on oeis.org

7059, 16566, 83127, 155265, 717816, 1336533, 4692864, 9375828, 17661567, 94178238, 177465387, 961030158, 1812060327, 9042662508, 17095324917, 89037683988, 177976357086, 858730036857, 1617360074715, 6792060711876

Offset: 0

Klaus Brockhaus, Jul 07 2001

			a(1) = 7059 + 9507 = 16566.

Reinhard Zumkeller and Michael Lee, Table of n, a(n) for n = 0..2432 (terms 0 through 250 provided by Reinhard Zumkeller)
Index entries for sequences related to Reverse and Add!

Cf. A033865, A023108, A006960, A063058, A056964, A004086.

m := 7059; stop := 25; c := 0; rev := int_reverse(m); while m <> rev and c < stop do inc(c); write(m," "); m := m + rev; rev := int_reverse(m); end;

a063057 n = a063057_list !! n
a063057_list = iterate a056964 7059 -- Reinhard Zumkeller, Sep 22 2011

NestList[# + FromDigits[Reverse[IntegerDigits[#]]]&, 7059, 40] (* Vincenzo Librandi, May 03 2014 *)

cp's OEIS Frontend

A072217 Consider the Reverse and Add! problem (cf. A001127); of all the n-digit numbers N which eventually reach a palindrome, pick that number N which takes the greatest number of steps to converge (in case of a tie, pick the smallest N); sequence gives number of steps N takes to converge.

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A072218 Consider the Reverse and Add! problem (cf. A001127); of all the n-digit numbers N which eventually reach a palindrome, pick that number N which takes the greatest number of steps to converge (in case of a tie, pick the smallest N); sequence gives palindrome that is reached.

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A243238 Table T(n,r) of terms in the reverse and add sequences of positive integers n read by antidiagonals.

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Maple

Mathematica

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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Mathematica

PARI

Extensions

A033650 Trajectory of 7 under map x --> x + (x-with-digits-reversed).

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Haskell

Maple

Mathematica

A033651 Trajectory of 9 under map x->x + (x-with-digits-reversed).

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Haskell

Maple

Mathematica

A033649 Trajectory of 5 under map x->x + (x-with-digits-reversed).

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Haskell

Maple

Mathematica

A063433 'Reverse and Add!' trajectory of 10577.

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ARIBAS