A065199 -id:A065199 - cp's OEIS frontend

A065319 30 'Reverse and Add' steps are needed to reach a palindrome.

10548, 11538, 12528, 13518, 14508, 20547, 21537, 22527, 23517, 24507, 30546, 31536, 32526, 33516, 34506, 40545, 40849, 41535, 41839, 42525, 42829, 43515, 43819, 44505, 44809, 44899, 45889, 46879, 47869, 48859, 49849, 50544, 50848

Offset: 1

Klaus Brockhaus, Oct 30 2001

The number of steps starts at 0, so palindromes (cf. A002113) are excluded. 30 is a record (cf. A065199) of the number of 'Reverse and Add' steps needed to reach a palindrome.

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

Cf. A002113, A065198, A065199, A065318.

lenQ[n_]:= Length[NestWhileList[# + FromDigits[Reverse[IntegerDigits[#]]]&, n, #! = FromDigits[Reverse[IntegerDigits[#]]]&, 1, 31]] == 31; Select[Range[10000], lenQ] (* Vincenzo Librandi, Sep 24 2013 *)

Changed offset from 0 to 1 by Vincenzo Librandi, Sep 24 2013

A340397 Complete sequence of numeric generators of delayed palindromes used to get 289-step delayed palindromes.

Original entry on oeis.org

105956506309091459564960, 53528258705040674282425, 31769674352025337585712, 53528258709000674282425, 53528298705040670282425, 53528298709000670282425, 53928258705040674282025, 53928258709000674282025, 53928298705040670282025, 31564654352025334636412

Offset: 1

Anton Stefanov, Jan 06 2021

The terms of this sequence are based on the terms of A326414 and are used to extend 288-step delayed palindromes up to 289-step delayed palindromes with the help of brute force extension of the method of undetermined coefficients.

			The 285-step delayed palindrome 105956506309091459564960 can be extended to the 286-step delayed palindrome 53528258705040674282425, which in turn can be extended to the 287-step delayed palindrome 31769674352025337585712. This 287-step solution is unextendable so we start a new branch from the 286-step solution 53528258709000674282425 which also extends the first term of our sequence, and so on.

Anton Stefanov, Table of n, a(n) for n = 1..20
Jason Doucette and Anton Stefanov, Final 289 steps World Record proof with some numeric generators of delayed palindromes inside
Anton Stefanov, Extending the method of undetermined coefficients
Anton Stefanov, isExtends test realisation on C# (to verify A340397 elements)

A326414 contains two numeric generators of delayed palindromes from our sequence: 16232852231012114813251 and 16232892231012110813251.
A072216 contains delayed palindromes of n digits with the greatest number of steps to converge.
A065199 contains records for the number of 'Reverse and Add' steps needed to reach a palindrome.
A065198 contains the most delayed palindromes in the order of their discovery and gives the starting points for the corresponding elements of the sequence A065199.

A353185 Numbers which require exactly 289 'Reverse and Add' steps to reach a palindrome.

Original entry on oeis.org

10037000230509917799950, 10037000240508917799950, 10037000250507917799950, 10037000260506917799950, 10037000270505917799950, 10037000280504917799950, 10037000290503917799950, 10037000330509817799950, 10037000340508817799950

Offset: 1

Andrey S. Shchebetov and Sergei D. Shchebetov, Apr 29 2022

The sequence starts with 10037000230509917799950, ends with 15999771990503200073000 and contains 9031680 terms known at present, including 13968441660506503386020 and 13568441660506503386420 discovered by Anton Stefanov on January 5, 2021.

Popular Computing (Calabasas, CA), The 196 Problem, Vol. 3 (No. 30, Sep 1975).

Sergei D. Shchebetov, Table of n, a(n) for n = 1..10000
Jason Doucette, World Records
Yutaka Nishiyama, Numerical Palindromes and the 196 Problem, International Journal of Pure and Applied Mathematics, Volume 80, No. 3, 2012, 375-384.
Sergei D. Shchebetov, 9031680 terms (zipped file)
R. Styer, The Palindromic Conjecture and the Fibonacci Sequence, Villanova University, 1986, 1-11.
C. W. Trigg, Palindromes by Addition, Mathematics Magazine, 40 (1967), 26-28.
C. W. Trigg, More on Palindromes by Reversal-Addition, Mathematics Magazine, 45 (1972), 184-186.
Wikipedia, Lychrel Number
196 and Other Lychrel Numbers, 196 and Lychrel Number
Index entries for sequences related to Reverse and Add!

A023109, A033672, A065198, A065199, A065320, A065321, A065322, A065323, A065324, A065325, A065326, A065327, A070743, A072216, A072217, A072218, A281301, A281390, A281506, A281507, A326414.

Each term requires exactly 289 steps to turn into a 142-digit palindrome.

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.

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A065319 30 'Reverse and Add' steps are needed to reach a palindrome.

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A340397 Complete sequence of numeric generators of delayed palindromes used to get 289-step delayed palindromes.

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A353185 Numbers which require exactly 289 'Reverse and Add' steps to reach a palindrome.

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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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