A281506 Numbers which require exactly 261 'Reverse and Add' steps to reach a palindrome.

1186060307891929990, 1186060317791929990, 1186060327691929990, 1186060337591929990, 1186060347491929990, 1186060357391929990, 1186060367291929990, 1186060377191929990, 1186060387091929990, 1186060407881929990, 1186060417781929990, 1186060427681929990, 1186060437581929990

Offset: 1

Author

Andrey S. Shchebetov and Sergei D. Shchebetov, Jan 23 2017

Keywords

• nonn
• base

Comments

The sequence starts with 1186060307891929990 (the 19-digit number also known as "the most delayed palindrome" and claimed as the world record, discovered by Jason Doucette on Nov 30 2005 and rediscovered by Vaughn Suite on Jan 02 2006) and continues for another 125 terms (none previously reported) each turning into a 119-digit palindrome after 261 steps until the sequence ends with 1186061987030929990. The distance between successive terms in the reported sequence has 9000000 as the greatest common divisor. No further numbers beyond 1186061987030929990 belonging to the same sequence are known, discovered or reported. The sequence was found empirically using computer modeling algorithms.

The sequence was extended to 108864 terms in total and ends with 1999291987030606810 - the last term of A281508 (see a-file). The sequence is complete - no further numbers beyond 1999291987030606810 belonging to the same sequence exist. The sequence was predicted theoretically and found empirically using computer modeling algorithms. - Sergei D. Shchebetov, May 12 2017

Comments from Sergei D. Shchebetov, Nov 14 2019: (Start)

There are two reasons that 1186060307891929990 cannot be the smallest term.

(1) Empirical: All numbers below were tested and none was found to have 261 (or higher) steps delay. This is presented, for example, in the Doucette link.

(2) Theoretical: There is no other combinations of the digits at 1186060307891929990 that gives you a lower number with the same reverse-and-sum result after the first step. This is because the number starts with 1 and you cannot go below 1 for the largest digit. Then it has 9s as the last 3 smallest digits and you cannot go up from there, but you could go down for the smallest digits (meaning up for the largest). For example, 1286060307891929980 (look at changes in the second digit from both ends: 1 turns into 2 and 9 turns into 8 with the sum staying 10 in both cases) would have the same 261-step delay. Same is with 1386060307891929970, etc. If you calculate all possible combinations where the pairwise sum of the digits stays the same, you will get 108864 terms.

Also, since 2005, when 1186060307891929990 was discovered, people have checked all numbers up to 23-digit range and found none (except for our set) with 261-step (or higher) delays. So finding a number with a 288-step delay, as Rob van Nobelen did, was a real breakthrough.

(End)

Examples

			Each term requires exactly 261 steps to turn into a 119-digit palindrome, the last term of A281507, and is separated by some multiples of 9000000 from the adjacent sequence terms.
		

References

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

Links

• Sergei D. Shchebetov, Table of n, a(n) for n = 1..126
• 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, 108864 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!

Crossrefs

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