A065199 -id:A065199 - cp's OEIS frontend

A281509 Trajectory of 1999291987030606810 (the largest presently known "most delayed palindrome") under the "Reverse and Add!" operation.

1999291987030606810, 2185352294922536801, 3271704589845072613, 6434410079699144336, 12768830049399288682, 41457129443403175403, 71914259877895350817, 143719619755790592734, 581014717313707510075, 1151030424627424920260, 1771324671891665221771, 3542550333873429453542, 5996099577656760005995

Offset: 0

Andrey S. Shchebetov and Sergei D. Shchebetov, Jan 24 2017

1999291987030606810 is the largest of the 126 presently known numbers that require exactly 261 steps to turn into a palindrome (see A281508). It is also the largest discovered "most delayed palindrome". The sequence reaches a 119-digit palindrome after 261 steps (see b-file). The number was obtained empirically using computer algorithms and was not reported before.

			a(1) = 1999291987030606810 + 186060307891929991 = 2185352294922536801.

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

Sergei D. Shchebetov, Table of n, a(n) for n = 0..261
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.
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!

Cf. A023109, A033672, A065198, A065199, A065320, A065321, A065322, A065323, A065324, A065325, A065326, A065327, A070743, A072216, A072217, A072218, A281301, A281390, A281506, A281507, A281508.

a(n+1) = a(n) + rev(a(n)).

a(n) = A281507(n) for n>0. - R. J. Mathar, Jan 27 2017

A306599 In base 8: a(n) sets a new record for the number of Reverse and Add! steps needed to reach a palindrome starting with a(n).

Original entry on oeis.org

0, 8, 15, 39, 47, 109, 375, 1591, 4470, 4596, 6393, 6718, 7930, 16439, 28903, 34292, 49465, 264496, 265750, 266126, 311290, 2067455, 16808764, 18276328, 1074296036, 1075478361

Offset: 0

A.H.M. Smeets, Feb 27 2019

It is conjectured that if a Reverse and Add! trajectory reaches a palindrome, it will be reached in relatively few steps, or otherwise it will never reach a palindrome. - A.H.M. Smeets, May 30 2019

Records in A306600.

Base 10: A065198 and A065199.

a(21)-a(25) from A.H.M. Smeets, May 30 2019

A306600 In base 8, records for the number of 'Reverse and Add' steps needed to reach a palindrome.

Original entry on oeis.org

0, 1, 2, 3, 6, 14, 17, 19, 21, 22, 23, 29, 30, 36, 46, 59, 60, 64, 67, 94, 95, 97, 120, 122, 133, 164

Offset: 0

A.H.M. Smeets, Feb 27 2019

It is conjectured that if a Reverse and Add! trajectory reaches a palindrome, it will be reached in relatively few steps, or otherwise it will never reach a palindrome.

For all n, all numbers m < A306599(n) will reach a palindrome within a(n) Reverse and Add! steps, or otherwise it is conjectured never to reach a palindrome.

Record setting numbers in A306599.

Base 10: A065198 and A065199.

a(21)-a(25) from A.H.M. Smeets, May 30 2019

A326414 Numbers which require exactly 288 'Reverse and Add' steps to reach a palindrome.

Original entry on oeis.org

12000700000025339936491, 12000700001015339936491, 12000700002005339936491, 12000700010024339936491, 12000700011014339936491, 12000700012004339936491, 12000700020023339936491, 12000700021013339936491, 12000700022003339936491, 12000700030022339936491

Offset: 1

Andrey S. Shchebetov and Sergei D. Shchebetov, Oct 18 2019

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, 19353600 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!

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

Each term requires exactly 288 steps to turn into a 142-digit palindrome.

Deleted an erroneous comment that said that the sequence was finite. - N. J. A. Sloane, Jun 23 2022

A070744 Records for the index of the (presumably) last palindrome in the 'Reverse and Add' trajectory of n.

Original entry on oeis.org

18, 32, 36, 37, 38, 39, 40, 54, 80, 82, 100, 101, 102

Offset: 1

Klaus Brockhaus, May 03 2002

Successive maxima in sequence A070742. A070743 gives the corresponding integers at which these records are attained.

Index entries for sequences related to Reverse and Add!

Cf. A065001, A065199, A070742, A070743.

Offset corrected by Sean A. Irvine, Jun 11 2024

A072147 Records for the length of the preperiodic part of the 'Reverse and Subtract' trajectories.

Original entry on oeis.org

0, 1, 2, 6, 7, 12, 13, 18, 25, 40, 45, 47, 48, 49, 55, 56, 60, 62, 63, 64, 66, 71, 72, 75, 78, 81, 106, 108, 111, 112, 114, 115, 119, 121, 122, 130, 132, 133, 135, 147, 148, 149, 151, 156

Offset: 1

Klaus Brockhaus, Jun 24 2002

Successive maxima in sequence A072137. A072146 gives the corresponding starting points. - This sequence is a weak analog of A065199, which uses 'Reverse and Add'.

			6 is a record, since the preperiodic part of the trajectory of 13 has length 6 and for k < 13 the preperiodic part has a smaller length (at most 2).

Cf. A072137, A072146, A065199.

a(18) inserted, a(25) corrected, a(29) through a(44) added by Alexander Pesch (alex-physics(AT)gmx.net), May 29 2007

Edited by N. J. A. Sloane, Dec 01 2007

A077407 In base 3, records for the number of Reverse and Add! steps needed to reach a palindrome.

Original entry on oeis.org

0, 1, 2, 3, 4, 18, 20, 21, 27, 28, 55, 56, 57, 70

Offset: 1

Klaus Brockhaus, Nov 05 2002

RECORDS transform of A077402. Base-3 analog of A066145 (base 2), A075687 (base 4) and A065199 (base 10). A077406 gives the corresponding starting points.

Probable Lychrel numbers are ignored, the first of which is 103. - Sean A. Irvine, Apr 19 2010

			Starting with 15, 3 Reverse and Add! steps are needed to reach a palindrome; starting with n < 15, less (at most 2) steps are needed.

N. J. A. Sloane, Transforms
Index entries for sequences related to Reverse and Add!

Cf. A065199, A066145, A075687, A077402, A077406.

a(14) from Sean A. Irvine, Apr 19 2010

A286481 Numbers which require exactly 260 'Reverse and Add' steps to reach a palindrome.

Original entry on oeis.org

1003062289999939142, 1003062299899939142, 1003062389989939142, 1003062399889939142, 1003062489979939142, 1003062499879939142, 1003062589969939142, 1003062599869939142, 1003062689959939142, 1003062699859939142, 1003062789949939142, 1003062799849939142, 1003062889939939142, 1003062899839939142, 1003062989929939142, 1003062999829939142

Offset: 1

Andrey S. Shchebetov and Sergei D. Shchebetov, May 12 2017

The sequence starts with 1003062289999939142 (the 19-digit number discovered by Vaughn Suite on Mar 19 2006) and continues for another 430079 terms (none previously reported) each turning into a 119-digit palindrome after 260 steps until the sequence ends with 3419399999822603000 (see a-file). No further numbers beyond 3419399999822603000 belonging to the same sequence exist. The sequence was predicted theoretically and found empirically using computer modeling algorithms. For the first 100 terms of the sequence see b-file.

			a(1) = 1003062289999939142 + 2419399999822603001 = 3422462289822542143

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

Sergei D. Shchebetov, Table of n, a(n) for n = 1..99
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, 430080 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!

Cf. A023109, A033672, A065198, A065199, A065320, A065321, A065322, A065323, A065324, A065325, A065326, A065327, A070743, A072216, A072217, A072218, A281301, A281390, A281506, A281507, A281508, A281509.

a(n+1) = a(n) + rev(a(n)).

A323975 a(n) sets a new record for the Lychrel number a(n) of 'Reverse and Add' steps, needed to reach a Lychrel number m < a(n) (i.e., its seed).

Original entry on oeis.org

195, 295, 4799, 10653, 10821, 14995, 105985, 1005523, 1008927, 1009413, 1029983

Offset: 1

A.H.M. Smeets, Feb 10 2019

Records in A323976.

Similar to the number of steps needed to reach a palindrome in the Reverse and Add! trajectories (see A065198 and A065199), the number of steps needed for a Lychrel number to reach the trajectory of its seed is relatively small.

Cf. A065198, A065199, A323976.

A323976 Records for the number of 'Reverse and Add' steps, needed for a Lychrel number to join the trajectory of a smaller Lychrel number (i.e., its seed).

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 10, 12, 14, 15, 19

Offset: 1

A.H.M. Smeets, Feb 10 2019

Similar to the number of steps needed to reach a palindrome in the Reverse and Add! trajectories (see A065198 and A065199), the number of steps needed for a Lychrel number to reach the trajectory of its seed is relatively small.

Cf. A065198, A065199, A323975.

cp's OEIS Frontend

A281509 Trajectory of 1999291987030606810 (the largest presently known "most delayed palindrome") under the "Reverse and Add!" operation.

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A306599 In base 8: a(n) sets a new record for the number of Reverse and Add! steps needed to reach a palindrome starting with a(n).

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A306600 In base 8, records for the number of 'Reverse and Add' steps needed to reach a palindrome.

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A326414 Numbers which require exactly 288 'Reverse and Add' steps to reach a palindrome.

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A070744 Records for the index of the (presumably) last palindrome in the 'Reverse and Add' trajectory of n.

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A072147 Records for the length of the preperiodic part of the 'Reverse and Subtract' trajectories.

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A077407 In base 3, records for the number of Reverse and Add! steps needed to reach a palindrome.

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Extensions

A286481 Numbers which require exactly 260 'Reverse and Add' steps to reach a palindrome.

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Formula

A323975 a(n) sets a new record for the Lychrel number a(n) of 'Reverse and Add' steps, needed to reach a Lychrel number m < a(n) (i.e., its seed).

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A323976 Records for the number of 'Reverse and Add' steps, needed for a Lychrel number to join the trajectory of a smaller Lychrel number (i.e., its seed).

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