A075252 -id:A075252 - cp's OEIS frontend

A092210 Positive integers n such that the trajectory of n under the Reverse and Add! operation carried out in base 2 (presumably) does not join the trajectory of any m < n.

1, 16, 64, 74, 98, 107, 259, 266, 271, 275, 298, 398, 442, 454, 522, 794, 911, 1027, 1046, 1057, 1066, 1070, 1073, 1076, 1081, 1082, 1085, 1115, 1274, 1422, 1562, 1570, 1967, 2070, 2329, 3106, 3355, 3871, 4099, 4114, 4120, 4126, 4136, 4152, 4182, 4189

Offset: 1

Klaus Brockhaus, Feb 25 2004

The conjecture that the base 2 trajectories of the terms do not join is based on the observation that if the trajectories of two integers < 12000 join, this happens after at most 93 steps, while for any two terms listed above the trajectories do not join within 1000 steps. For pairs from 1, 16, 64, 74, 98, 107 this has even been checked for 5000 steps.

Base-2 analog of A070788 (base 10) and A091675 (base 4).

			The trajectory of 2 is part of the trajectory of 1 (cf. A035522); the trajectory of 16 does not join the trajectory of 1 within 10000 steps; the trajectory of 64 does not join the trajectory of 1 or of 16 within 10000 steps.

Index entries for sequences related to Reverse and Add!

Cf. A035522, A075252, A070788, A091675.

limit = 10^3; (* Assumes that there is no palindrome if none is found before "limit" iterations *)
utraj = NestList[# + IntegerReverse[#, 2] &, 1, limit];
Flatten@{1, Select[Range[2, 4189],   (l = Length@NestWhileList[# + IntegerReverse[#, 2] &, #, ! MemberQ[utraj, #] &, 1, limit];
  utraj = Union[utraj, NestList[# + IntegerReverse[#, 2] &, #, limit]];
  l == limit + 1) &]} (* Robert Price, Nov 03 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).

A092212 a(n) = smallest non-palindromic k such that the base-2 Reverse and Add! trajectory of k is palindrome-free and joins the trajectory of A092210(n).

Original entry on oeis.org

26, 65649, 89, 4193, 3599, 775, 68076, 2173

Offset: 1

Klaus Brockhaus, Feb 25 2004

Terms a(9) to a(29) are 205796147 (conjectured), 4402, 16720, 1089448, 442, 537, unknown, 1050177, 1575, 28822, unknown, 40573, 1066, 1587, unknown, unknown, 1081, 1082, 1085, 1115, 4185.
a(n) >= A092210(n); a(n) = A092210(n) iff the trajectory of A092210(n) is palindrome-free, i.e., A092210(n) is also a term of A075252.
a(n) determines a 1-to-1 mapping from the terms of A092210 to the terms of A075252, the inverse of the mapping determined by A092211.
The 1-to-1 property of the mapping depends on the conjecture that the base-2 Reverse and Add! trajectory of each term of A092210 contains only a finite number of palindromes (cf. A092215).
Base-2 analog of A089494 (base 10) and A091677 (base 4).

			A092210(3) = 64, the trajectory of 64 joins the trajectory of 89 at 48480, so a(3) = 89. A092210(5) = 98, the trajectory of 98 joins the trajectory of 3599 = A075252(16) at 401104704, so a(5) = 3599.

Index entries for sequences related to Reverse and Add!

Cf. A075252, A092210, A092211, A092215, A089494, A091677.

limit = 10^3; (* Assumes that there is no palindrome if none is found before "limit" iterations *)
utraj = NestList[# + IntegerReverse[#, 2] &, 1, limit];
A092210 = Flatten@{1, Select[Range[2, 266], (l =
        Length@NestWhileList[# + IntegerReverse[#, 2] &, #, !
            MemberQ[utraj, #] &, 1, limit];
       utraj =
        Union[utraj, NestList[# + IntegerReverse[#, 2] &, #, limit]];
       l == limit + 1) &]};
A092212 = {};
For[i = 1, i <= Length@A092210, i++,
k = A092210[[i]];
itraj = NestList[# + IntegerReverse[#, 2] &, A092210[[i]], limit];
While[ktraj =
   NestWhileList[# + IntegerReverse[#, 2] &,
    k, # != IntegerReverse[#, 2] &, 1, limit];
  PalindromeQ[k] || Length@ktraj != limit + 1 || ! IntersectingQ[itraj, ktraj], k++];
AppendTo[A092212, k]]; A092212 (* Robert Price, Nov 03 2019 *)

a(1) and a(3) corrected by Robert Price, Nov 06 2019

A306481 Lychrel numbers k that set a new record for the number of 'Reverse and Add' steps in base 2 needed to reach a Lychrel number m < k (i.e., its seed).

Original entry on oeis.org

22, 26, 67, 106, 157, 199, 281, 1145, 1594, 1798, 4326, 12794, 17407, 18428, 67583, 69628, 73978

Offset: 1

A.H.M. Smeets, Feb 18 2019

Records in A306482.
Similar to the number of steps needed to reach a palindrome in the Reverse and Add! trajectories (see A066144 and A066145), the number of steps needed for a Lychrel number to reach the trajectory of its seed is relatively small.
Lychrel numbers in A066059; seeds in A075252 (for base 2).
As a clarification, this sequence can also be described as: Base 2 Lychrel numbers (A066059) k that sets a new record for the number of 'Reverse and Add' steps in base 2 needed to reach the trajectory of a base 2 Lychrel number seed (A075252) that is less than k. - Robert Price, Nov 20 2019

Cf. A066059, A066144, A066145, A075252, A306482.

limit = 200; (* Assumes that there is no palindrome if none is found before "limit" iterations *)
A066059 = Select[Range[50000],
   Length@NestWhileList[# + IntegerReverse[#, 2] &, #, # !=
         IntegerReverse[#, 2] &, 1, limit] == limit + 1 &];
utraj = {};
A075252 = Select[Range[50000],
   (x = NestWhileList[# + IntegerReverse[#, 2] &, #, # !=
         IntegerReverse[#, 2] & , 1, limit];
     If[Length@x >= limit  && Intersection[x, utraj] == {},
      utraj = Union[utraj, x]; True,
      utraj = Union[utraj, x]]) &];
A306481 = {}; best = -1; lastj = 0;
utraj = {};
For[i = 1, i <= Length@A066059, i++,
For[j = lastj + 1, j <= Length@A075252, j++,
  If[A066059[[i]] < A075252[[j]], Break[]];
  utraj = Union[utraj, NestList[# + IntegerReverse[#, 2] &, A075252[[j]], limit]];
  lastj = j; ];
l = NestWhileList[# + IntegerReverse[#, 2] &,
   A066059[[i]], ! MemberQ[utraj, #] &, 1, limit];
If[Length@l == limit + 1, Continue[]];
If[Length@l > best, best = Length@l; AppendTo[A306481, A066059[[i]]]];
]; A306481 (* Robert Price, Nov 20 2019 *)

A306482 Records for the number of 'Reverse and Add' steps in base 2 needed for a Lychrel number to join the trajectory of a smaller Lychrel number (i.e., its seed). Record setting numbers in A306481.

Original entry on oeis.org

0, 4, 5, 6, 9, 11, 17, 21, 22, 34, 52, 68, 83, 84, 91, 92, 98

Offset: 1

A.H.M. Smeets, Feb 18 2019

Record setting numbers in A306481.
Similar to the number of steps needed to reach a palindrome in the Reverse and Add! trajectories (see A066144 and A066145), the number of steps needed for a Lychrel number to reach the trajectory of its seed is relatively small.
Lychrel numbers in A066059; seeds in A075252 (for base 2).
As a clarification, this sequence can also be described as: "Records for the number of 'Reverse and Add' steps in base 2 needed for a base 2 Lychrel number (A066059) to join the trajectory of a smaller base 2 Lychrel number seed (A075252)." - Robert Price, Nov 20 2019

Cf. A066059, A066144, A066145, A075252, A306481.

limit = 200; (* Assumes that there is no palindrome if none is found before "limit" iterations *)
A066059 = Select[Range[50000],
   Length@NestWhileList[# + IntegerReverse[#, 2] &, #, # !=
         IntegerReverse[#, 2] &, 1, limit] == limit + 1 &];utraj = {};
A075252 = Select[Range[50000],    (x = NestWhileList[# + IntegerReverse[#, 2] &, #, # !=
         IntegerReverse[#, 2] & , 1, limit];
     If[Length@x >= limit  && Intersection[x, utraj] == {},
      utraj = Union[utraj, x]; True,
      utraj = Union[utraj, x]]) &];A306482 = {}; best = -1; lastj = 0;
utraj = {};
For[i = 1, i <= Length@A066059, i++,
 For[j = lastj + 1, j <= Length@A075252, j++,
  If[A066059[[i]] < A075252[[j]], Break[]];
  utraj = Union[utraj, NestList[# + IntegerReverse[#, 2] &, A075252[[j]], limit]];
  lastj = j; ];
 l = NestWhileList[# + IntegerReverse[#, 2] &,
   A066059[[i]], ! MemberQ[utraj, #] &, 1, limit];
 If[Length@l == limit + 1, Continue[]];
If[Length@l > best, best = Length@l; AppendTo[A306482, Length@l - 1]]; ]; A306482  (* Robert Price, Nov 20 2019 *)

A092215 Smallest number whose base-2 Reverse and Add! trajectory (presumably) contains exactly n base-2 palindromes, or -1 if there is no such number.

Original entry on oeis.org

22, 30, 10, 4, 6, 2, 1, 132, 314, 403, 259, 2048, -1, -1, -1, -1

Offset: 0

Klaus Brockhaus, Feb 25 2004

Conjecture 1: For each k > 0 the trajectory of k eventually leads to a term in the trajectory of some j which belongs to A075252, i.e., whose trajectory (presumably) never leads to a palindrome. Conjecture 2: There is no k > 0 such that the trajectory of k contains more than eleven base 2 palindromes, i.e., a(n) = -1 for n > 11.

Base-2 analog of A077594 (base 10) and A091680 (base 4).

			a(4) = 6 since the trajectory of 6 contains the four palindromes 9, 27, 255, 765 (1001, 11011, 11111111, 1011111101 in base 2) and at 48960 joins the trajectory of 22 = A075252(1) and the trajectories of 1 (A035522), 2, 3, 4, 5 contain resp. 6, 5, 5, 3, 3 palindromes.

Index entries for sequences related to Reverse and Add!

Cf. A035522, A006995, A075252, A077594, A091680.

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

A092210 Positive integers n such that the trajectory of n under the Reverse and Add! operation carried out in base 2 (presumably) does not join the trajectory of any m < n.

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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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A092212 a(n) = smallest non-palindromic k such that the base-2 Reverse and Add! trajectory of k is palindrome-free and joins the trajectory of A092210(n).

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A306481 Lychrel numbers k that set a new record for the number of 'Reverse and Add' steps in base 2 needed to reach a Lychrel number m < k (i.e., its seed).

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A306482 Records for the number of 'Reverse and Add' steps in base 2 needed for a Lychrel number to join the trajectory of a smaller Lychrel number (i.e., its seed). Record setting numbers in A306481.

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A092215 Smallest number whose base-2 Reverse and Add! trajectory (presumably) contains exactly n base-2 palindromes, or -1 if there is no such number.

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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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