A089381 L-th order palindromes with L > 2.

10917, 11907, 11997, 12987, 13977, 14967, 15957, 16947, 17937, 18927, 19917, 20997, 21834, 21987, 22977, 23814, 23967, 23994, 24957, 25497, 25947, 25974, 26487, 26937, 27477, 27927, 27954, 28467, 28917, 29457, 29907, 29934, 30915

Offset: 1

Author

Darrell Plank (jar_czar(AT)msn.com), Dec 28 2003

Keywords

• base
• nonn

Comments

Let P(m) = m/2 if m is even, m + rev(m) if m is odd, where rev(m) is m's base 10 representation reversed. It is conjectured that any number k eventually cycles when P is repeatedly applied to it. If the cycle has length L, k is called an L-th order palindrome.

It has not been proved that every number eventually cycles, but all numbers less than a million do. Palindromes of order L > 2 seem to be quite rare. 10917 is the smallest and has order 7. There are 263 less than 100000 and 7745 less than 1000000.

The first number with L > 2 that doesn't end in the same cycle as 10917 is 1000353. Other cycles are known, most of them fairly small, but one has length 327 (starting with 1447132589595).

There are an infinite number of different cycles of length 7 because one can insert any number of 9's in the middle of a number in the 7th-order cycle and get a new cycle of length 7 - e.g., taking the number 13748625 from the cycle, one can produce another cycle from 13749998625.

I believe this is not a straightforward generalization of ordinary palindromes (A002113) - they are not the same as 2nd-order palindromes. - N. J. A. Sloane, Jan 01 2004

Examples

			For most numbers, iterating P produces a cycle of length 2: e.g., 121 -> 242 -> 121 -> ...
The sequence for 10917 is 10917, 82818, 41409, 131823, 459954, 229977, 1009899, 10998900, 5499450, 2749725, 8029197, 15948405, {66433356, 33216678, 16608339, 109989000, 54994500, 27497250, 13748625} where the numbers in the brackets repeat. There are 7 numbers inside the brackets so 10917 is a 7th-order palindrome.
		

References

• C. A. Pickover, Wonders of Numbers, Adventures in Mathematics, Mind and Meaning, Chapter 58, 'Emordnilap Numbers,' Oxford University Press, N.Y., 2001, pp. 142-144.

Crossrefs

Cf. A089605, A006960, A023108, A002113.

Programs

Mathematica

Step[n_] := If[ EvenQ[n], n/2, n + FromDigits[ Reverse[ IntegerDigits[n]]]]; cPalHash = 1013; clearArray = Array[{} &, cPalHash]; InsertCheck[n_, a_] := Module[{i = Mod[n, cPalHash] + 1}, a[[i]] = Append[ a[[i]], n]]; SetAttributes[ InsertCheck, HoldRest]; CheckArray[n_, a_] := MemberQ[ a[[Mod[n, cPalHash] + 1]], n]; SetAttributes[ CheckArray, HoldRest]; PalListHelper[n_, cTries_] := Module[ {ch = clearArray}, NestWhileList[ (InsertCheck[ #, ch]; Step[ # ]) &, n, Not[CheckArray[ #, ch]] &, 1, cTries]]; PalList[n_, cTries_] := Module[ {lst, nRemoved, loop}, lst = PalListHelper[n, cTries]; nRemoved = First[ First[ Position[ lst, lst[[ -1]]]]]; loop = Drop[ Take[ lst, {nRemoved, -1}], -1]; Append[ Take[ lst, {1, nRemoved - 1}], loop]]; Select[ Range[ 31000], Length[ PalList[ #, 1013][[ -1]]] > 2 &]

Extensions

Edited by Robert G. Wilson v and N. J. A. Sloane, Dec 31 2003