A077405 -id:A077405 - cp's OEIS frontend

A063048 Numbers n such that the Reverse and Add! trajectory of n (presumably) does not reach a palindrome and does not join the trajectory of any term m < n.

196, 879, 1997, 7059, 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

Offset: 1

Klaus Brockhaus, Jul 07 2001, revised Nov 04 2003

The starting number n is regarded as part of the trajectory, so palindromes are excluded from the sequence. A088753 is obtained if palindromes are not excluded. The smallest term in A063048 but not in A088753 is 19098, the smallest term in A088753 but not in A063048 is 9999.
Subsequence of A023108. Sequence A070788 is similarly defined, but palindromes are irrelevant. Corresponding sequences for other bases are A075252 (base 2), A077405 (base 3), A075421 (base 4).
If the trajectory of a number k joins the trajectory of a smaller number which is a term of the present sequence, then this occurs after very few Reverse and Add! steps (at most 8 for k < 100000, at most 10 for k < 1000000). On the other hand, the trajectories of the terms < 14000 do not join the trajectory of any smaller term within at least 1500 steps. This is the precise meaning of "presumably" in the definition.
The terms are rather unevenly distributed. They form clusters, especially above 10^4, 10^5, 10^6, ... . The interval from 10000 to 11000 for example contains 26 terms, whereas only two terms occur in the interval from 90000 to 100000.
It seems that if the most significant digit is not equal to 1, the least significant digit is always 9, while this does not hold for the Lychrel numbers as in A023108. - A.H.M. Smeets, Feb 18 2019
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".
The plot log_10(a(n)) versus log_10(n) shows a stepwise behavior. However, the global behavior seems to be a straight line with slope e/(e-1) (= A185393). This slope is also obtained for the seeds in the Reverse and Add! problem in other bases. (End)

			1997 is a term since the trajectory of 1997 (presumably) does not lead to a number which occurs in the trajectory of 196 or of 879 (actually checked for the first 10000 terms of these trajectories). The trajectory of 1997 joins the trajectory of 106 at 97768 (cf. A070796), but 106 is not a term of the present sequence.

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..21491 (terms 1..1711 from Klaus Brockhaus).
Klaus Brockhaus, Illustration: Distribution of terms below 2000000
Klaus Brockhaus, List of terms below 2000000
A.H.M. Smeets, log_10(a(n)) versus log_10(n)
Index entries for sequences related to Reverse and Add!

Cf. A033865, A023108, A006960, A063049 - A063065, A088753, A070788, A075252, A077405, A075421, A070796.

limit = 10^3; (* Assumes that there is no palindrome if none is found before "limit" iterations *)
utraj = {};
Select[Range[0, 14000], (x = NestWhileList[ # + IntegerReverse[#] &, #, ! PalindromeQ[#] &, 1, limit];
   If[Length[x] >= limit && Intersection[x, utraj] == {},
    utraj = Union[utraj, x]; True,
utraj = Union[utraj, x]]) &] (* Robert Price, Oct 16 2019 *)

A077404 Trajectory of n under the Reverse and Add! operation carried out in base 3 (presumably) does not reach a palindrome.

Original entry on oeis.org

103, 127, 167, 176, 180, 183, 185, 191, 197, 200, 206, 207, 209, 215, 221, 230, 239, 291, 309, 318, 340, 358, 369, 376, 387, 394, 396, 413, 418, 431, 436, 449, 454, 467, 472, 582, 600, 618, 636, 655, 660, 673, 678, 691, 696, 709, 714, 746, 776, 803, 805

Offset: 1

Klaus Brockhaus, Nov 05 2002

Base-3 analog of A066059 (base 2), A075420 (base 4) and A023108 (base 10).

A.H.M. Smeets, Table of n, a(n) for n = 1..20000
Index entries for sequences related to Reverse and Add!

Cf. A023108, A066059, A075420, A077405.

m := 900; stop := 1000; for n := 0 to m do c := 0; k := n; b := 1; while b do d := k; rev := 0; while d > 0 do rev := 3*rev + (d mod 3); d := d div 3; end; if k = rev or c = stop then b := 0; else k := k + rev; end; inc(c); end; if c > stop then write(n,","); end; end;

limit = 10^3; (* Assumes that there is no palindrome if none is found before "limit" iterations *)
Select[Range[805],
 Length@NestWhileList[# + IntegerReverse[#, 3] &, #, # !=
IntegerReverse[#, 3]  &, 1, limit] == limit + 1 &] (* Robert Price, Oct 19 2019 *)

Offset changed to 1 by A.H.M. Smeets, Feb 10 2019

A306596 Trajectory of n under the Reverse and Add! operation carried out in base 8 (presumably) does not reach a palindrome and (presumably) does not join the trajectory of any term m < n.

Original entry on oeis.org

1021, 3623, 4327, 4382, 4404, 4413, 4444, 4500, 4502, 4518, 4522, 4528, 4530, 4575, 4592, 4594, 5117, 5502, 5629, 6270, 7806, 8247, 8607, 12503, 12527, 12535, 16319, 16383, 16815, 20711, 20975, 24751, 25015, 28351, 28415, 28671, 28775, 28791, 33757, 33766, 34254, 34286, 34757, 34781, 35268, 35276

Offset: 1

A.H.M. Smeets, Feb 27 2019

A number is considered here (presumably) a Lychrel number in base 8 if it does not reach a palindrome within 100 steps more than the actual record. For those record numbers of steps, see A306600; for the corresponding record-setting numbers, see A306599. Futhermore, a Lychrel number is considered not to reach the trajectory of any smaller Lychrel number if it does not reach a trajectory of a smaller Lychrel number within 100 steps more than the actual record. For those record number of steps see A306851, and its corresponding record setting numbers, see A306850.
For a(11) = 4522 we obtain a cyclic structure of the terms in its trajectory (starting at the 12th term in the trajectory) which can be represented by the context-free grammar with alphabet = {0,1,2,3,4,5,6,7} and production rules:
  S -> S_a | S_b | S_c | S_d | S_e | S_f | S_g | S_h,
  S_a -> 10 T_a  00, T_a -> 7 T_a 0 | 777670,
  S_b -> 11 T_b  01, T_b -> 0 T_b 7 | 076667,
  S_c -> 22 T_c  12, T_c -> 0 T_c 7 | 065557,
  S_d -> 44 T_d  34, T_d -> 0 T_d 7 | 043337,
  S_e -> 10 T_e 000, T_e -> 7 T_e 0 | 777670,
  S_f -> 11 T_f 701, T_f -> 0 T_f 7 | 007567,
  S_g -> 22 T_g 712, T_g -> 0 T_g 7 | 006357,
  S_h -> 44 T_h 734, T_h -> 0 T_h 7 | 003737;
i.e., the cycle length is 8.
For all other terms up to and including a(649) = 527823, no such structure has been obtained.

Cf. A306600, A306599, A306851, A306850.

Base-8 analog of A075252 (base 2), A077405 (base 3), A075421 (base 4) and A063048 (base 10).

A077402 Reverse and Add! carried out in base 3; number of steps needed to reach a palindrome, or -1 if no palindrome is ever reached.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 1, 2, 0, 1, 0, 2, 1, 0, 2, 3, 0, 4, 1, 2, 0, 1, 2, 0, 3, 4, 0, 1, 0, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 0, 2, 3, 3, 2, 1, 1, 2, 3, 3, 2, 3, 0, 18, 1, 2, 0, 1, 2, 4, 1, 2, 2, 1, 2, 4, 1, 2, 0, 3, 2, 4, 1, 2, 2, 3, 2, 4, 17, 18, 0, 1, 0, 2, 1, 1, 2, 1, 1, 3, 1, 0, 2, 1, 1, 16, 1, 1, 2, 2, 0, 2, 4, -1, 16, 3, 15, 2, 1, 1, 2, 1, 0, 3, 3, 3, 2, 1, 1, 16, 1

Offset: 0

Klaus Brockhaus, Nov 05 2002

Base-3 analog of A066057 (base 2), A075685 (base 4) and A033665 (base 10). a(103) = -1 is a conjecture (cf. A066450, A077408). For values of n such that presumably a(n) = -1 see A077404.

			17 (decimal) = 122 -> 122 + 221 = 1120 -> 1120 + 211 = 2101 -> 2101 + 1012 = 10120 -> 10120 + 2101 = 12221 (palindrome) = 160 (decimal) requires 4 steps, so a(17) = 4.

Index entries for sequences related to Reverse and Add!

Cf. A014190, A066450, A033665, A066057, A075685, A077404, A077405.

m := 120; stop := 1000; for n := 0 to m do v := -1; c := 0; k := n; while c < stop do d := k; rev := 0; while d > 0 do rev := 3*rev + (d mod 3); d := d div 3; end; if k = rev then v := c; c := stop; else inc(c); k := k + rev; end; end; write(v,","); end;

A077408 Trajectory of 103 under the Reverse and Add! operation carried out in base 3, written in base 10.

Original entry on oeis.org

103, 230, 436, 776, 2424, 3856, 7400, 20856, 30928, 60920, 220248, 242704, 432896, 857152, 1460408, 2754688, 5134016, 16206744, 24437488, 44623424, 138104472, 201737128, 401511824, 1438324704, 1601682040, 2820726320, 5622321088

Offset: 0

Klaus Brockhaus, Nov 05 2002

103 = A077405(0) is conjectured (cf. A066450) to be the smallest number such that the Reverse and Add! algorithm in base 3 does not lead to a palindrome. Its trajectory does not exhibit any recognizable regularity, so that the method by which the base-2 trajectories of 22 (cf. A061561), 77 (cf. A075253), 442 (cf. A075268) etc. as well as the base-4 trajectories of 318 (cf. A075153), 266718 (cf. A075466), 270798 (cf. A075467) etc. can be proved to be palindrome-free (cf. Links), is not applicable here.

			103 (decimal) = 10211 -> 10211 + 11201 = 22112 = 230 (decimal).

Index entries for sequences related to Reverse and Add!
Klaus Brockhaus, On the 'Reverse and Add!' algorithm in base 2

Cf. A058042, A077405, A066450, A061561, A075253, A075268, A075153, A075466, A075467.

m := 103; stop := 28; c := 0; while c < stop do write(m:group(0),","); k := m; rev := 0; while k > 0 do rev := 3*rev + (k mod 3); k := k div 3; end; inc(c); m := m+rev; end;

A344119 Numbers k whose trajectory under the Reverse and Add! operation carried out in base 16 does not reach a palindrome and (presumably) does not join the trajectory of any term m < k.

Original entry on oeis.org

413, 429, 443, 445, 3407, 3647, 3711, 3775, 3807, 3839, 4287, 7417, 12463, 12527, 16383, 24575, 28879, 45183, 45231, 49151, 57343, 61615, 61663, 61679, 66511, 66783, 67023, 67535, 67551, 67628, 67630, 67644, 67646, 67673, 67674, 67676, 67688, 67718, 67734

Offset: 1

A.H.M. Smeets, Aug 16 2021

A number is considered here (presumably) a Lychrel number in base 16 if it does not reach a palindrome within 200 steps more than the actual record. Those record numbers of steps to become palindromic are known from data in other bases not to increase that much (see for instance A065198 and A065199 in case of base 10). Furthermore, a Lychrel number is considered not to reach the trajectory of any smaller Lychrel number if it does not reach a trajectory of a smaller Lychrel number within 100 steps more than the actual record. Again, those record numbers of steps to reach the trajectory of a smaller Lychrel number are known from data in other bases not to increase that much (see for instance A323975 and A323976 in case of base 10).

In other bases: A075252 (2), A077405 (3), A075421 (4), A306596 (8), A063048 (10).

Cf. A065198, A065199, A323975, A323976.

cp's OEIS Frontend

A063048 Numbers n such that the Reverse and Add! trajectory of n (presumably) does not reach a palindrome and does not join the trajectory of any term m < n.

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A077404 Trajectory of n under the Reverse and Add! operation carried out in base 3 (presumably) does not reach a palindrome.

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A306596 Trajectory of n under the Reverse and Add! operation carried out in base 8 (presumably) does not reach a palindrome and (presumably) does not join the trajectory of any term m < n.

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A077402 Reverse and Add! carried out in base 3; number of steps needed to reach a palindrome, or -1 if no palindrome is ever reached.

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A077408 Trajectory of 103 under the Reverse and Add! operation carried out in base 3, written in base 10.

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A344119 Numbers k whose trajectory under the Reverse and Add! operation carried out in base 16 does not reach a palindrome and (presumably) does not join the trajectory of any term m < k.

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