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Since we do not even know if 196 eventually converges (see A006960, A023108) for n >= 3 these values are only conjectures.

Since we do not even know if 196 eventually converges (see A006960, A023108) for n >= 3 these values are only conjectures.

1000004999700144385 is the largest of the first 225 numbers that require exactly 259 steps to turn into a palindrome (see A281390). The sequence reaches a 119-digit palindrome after 259 steps (see b-file). The number was obtained empirically using computer algorithms and was not reported before.

Row 1000004999700144385 of the array in A243238. - Felix Fröhlich, Jan 21 2017

The sequence starts with 1000000079994144385 (the 19-digit number discovered by Vaughn Suite on Jul 26 2005 and rediscovered by Jason Doucette on Nov 28 2005) and continues for another 224 terms (none previously reported) each turning into a 119-digit palindrome after 259 steps until the sequence ends with 1000004999700144385. The distance between successive terms in the reported sequence has 9000000 as the greatest common divisor. No further numbers beyond 1000004999700144385 belonging to the same sequence are known, discovered or reported. The sequence was found empirically using computer modeling algorithms.

The sequence was extended to 1620000 terms in total and currently ends with 6834414999700000000 (see a-file). The sequence is complete - no further numbers beyond 6834414999700000000 belonging to the same sequence exist. The sequence was predicted theoretically and found empirically using computer modeling algorithms. - Sergei D. Shchebetov, May 12 2017

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)

1186061987030929990 is the largest of the first 126 numbers that require exactly 261 steps to turn into a palindrome (see A281506). 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.

The sequence starts with 1999290307891606810 and continues for another 125 terms (none previously reported, including the first term) each turning into a 119-digit palindrome after 261 steps until the sequence ends with 1999291987030606810. The distance between successive terms in the reported sequence has 9000000 as the greatest common divisor. No further numbers beyond 1999291987030606810 belonging to the same sequence are known, discovered or reported. Moreover, 1999291987030606810 is currently the largest discovered "most delayed palindrome". The sequence was found empirically using computer modeling algorithms.

It is only a conjecture that there are no further terms. - N. J. A. Sloane, 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.

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.