A066450 a(n) is the conjectured value of the minimal number to which repeated application of the "reverse and add!" algorithm in base n does not terminate in a palindrome. If there is no such number in base n, then a(n) := -1.

22, 103, 290, 708, 1079, 2656, 1021, 593, 196, 1011, 237, 2701, 361, 447, 413, 3297, 519, 341, 379, 711, 461, 505, 551, 1022, 649, 701, 755, 811, 869, 929, 991, 1055, 1799, 1922, 1259, 1331, 1405, 1481, 1559, 1639, 1595, 1762, 1891, 1934, 2069, 2161

Offset: 2

Author

Frederick Magata (frederick.magata(AT)uni-muenster.de), Dec 29 2001

Keywords

• nonn
• base

Comments

It would be nice to remove the word "Conjectured" from the description. - N. J. A. Sloane

All the terms in this sequence except the first are only conjectures. (See Walker, Irvin on a(10)=196 and Brockhaus on a(2)=22.)

An obvious algorithm is: start with r := n and check whether the "reverse and add!" algorithm in base n halts in a palindrome or not. If it stops, increment r by one and repeat the process, else return r. To obtain the values above, an upper limit of 100 "reverse and add!" steps was used.

Conjectures: a(n) shows the same asymptotic behavior as n^2. For infinitely many n, a(n) = n^2 - n - 1. Again, it is an open question, if the values of the sequence really lead to infinitely many "reverse and add!" steps or not. Is the sequence always positive?

Links

• Klaus Brockhaus, On the 'Reverse and Add!' algorithm in base 2
• T. Irvin, About Two Months of Computing, or An Addendum to Mr. Walker's Three Years of Computing.
• J. Walker, Three Years Of Computing: Final Report On The Palindrome Quest
• Index entries for sequences related to Reverse and Add!

Programs

Mathematica

limit = 10^3; (* Assumes that there is no palindrome if none is found before "limit" iterations *)
Table[SelectFirst[Range[10000],
  Length@NestWhileList[ # + IntegerReverse[#, n] &,  #, # !=
        IntegerReverse[#, n]  &, 1, limit] == limit + 1 &] , {n, 2,
47}] (* Robert Price, Oct 18 2019 *)

Extensions

David W. Wilson remarks (Jan 02 2002): I verified these using 1000 digits as a stopping point (this would be >>1000 iterations). I am highly confident of these values.