A066059 -id:A066059 - cp's OEIS frontend

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).

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 *)

A213012 Trajectory of 26 under the Reverse and Add! operation carried out in base 2.

Original entry on oeis.org

26, 37, 78, 135, 360, 405, 744, 837, 1488, 1581, 3024, 3213, 6048, 6237, 12192, 12573, 24384, 24765, 48960, 49725, 97920, 98685, 196224, 197757, 392448, 393981, 785664, 788733, 1571328, 1574397, 3144192, 3150333

Offset: 0

Ben Branman, Jun 01 2012

26 is the second-smallest number (after 22) whose base 2 trajectory does not contain a palindrome.
lim_{n -> infinity} a(n)/a(n-1) = 2 for n mod 2 = 0.
lim_{n -> infinity} a(n)/a(n-1) = 1 for n mod 2 = 1. - Branman
In 2001, Brockhaus proved that if the binary Reverse and Add trajectory of an integer contains an integer of one of four specific given  forms, then the trajectory never reaches a palindrome. In the case of 26, that would be 3(2^(2k + 1) - 2^k), with k = 3 corresponding to 360. - Alonso del Arte, Jun 02 2012

			In binary, 26 is 11010.
a(1) = 37 because 11010 + 01011 = 100101, or 37.
a(2) = 78 because 100101 + 101001 = 1001110, or 78.

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

Cf. A035522, A061561, A066059, A077076, A077077.

binRA[n_] := If[Reverse[IntegerDigits[n, 2]] == IntegerDigits[n, 2], n, FromDigits[Reverse[IntegerDigits[n, 2]], 2] + n]; NestList[binRA, 26, 100]

A306365 Trajectory of n under the Reverse and Add! operation carried out in base 5 (presumably) does not reach a palindrome.

Original entry on oeis.org

708, 718, 723, 731, 733, 743, 828, 838, 843, 851, 853, 863, 958, 963, 983, 1078, 1083, 1103, 1203, 1299, 1309, 1332, 1342, 1347, 1350, 1355, 1357, 1359, 1367, 1419, 1429, 1452, 1462, 1467, 1475, 1477, 1479, 1487, 1499, 1539, 1582, 1607, 1619, 1659, 1702, 1707, 1727, 1739, 1779, 1827, 1859, 1923, 1933, 1956

Offset: 1

A.H.M. Smeets, Feb 10 2019

Base-5 analog of A066059 (base 2), A077404 (base 3), A075420 (base 4) and A023108 (base 10).

All terms are tested up to 200 iteration steps, i.e., within 200 steps no palindrome was reached.

A.H.M. Smeets, Table of n, a(n) for n = 1..20000

Cf. A066059, A075420, A077404, A023108.

cp's OEIS Frontend

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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A213012 Trajectory of 26 under the Reverse and Add! operation carried out in base 2.

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A306365 Trajectory of n under the Reverse and Add! operation carried out in base 5 (presumably) does not reach a palindrome.

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