%I #76 Dec 23 2021 09:36:00
%S 22,103,290,708,1079,2656,1021,593,196,1011,237,2701,361,447,413,3297,
%T 519,341,379,711,461,505,551,1022,649,701,755,811,869,929,991,1055,
%U 1799,1922,1259,1331,1405,1481,1559,1639,1595,1762,1891,1934,2069,2161
%N In base n, a(n) is the smallest number m that leads to a palindrome-free sequence, using the following process: start with m; reverse the digits and add it to m, repeat. Stop if you reach a palindrome.
%C Only a(2) is proved, all the others are conjectured. - _Eric Chen_, Apr 20 2015 [corrected by _A.H.M. Smeets_, May 27 2019]
%C Brown's link reports a(3) as 103 instead of 100. What is the correct value? _Dmitry Kamenetsky_, Mar 06 2017 [a(3) = 103 is correct as from A077404, _A.H.M. Smeets_, May 27 2019]
%C From _A.H.M. Smeets_, May 27 2019: (Start)
%C It seems that a(n) < n^2 (i.e., a(n) in base n has two digits) and the least significant digit of a(n) in base n equals n-1, for n > 73.
%C For n <= 73 and the least significant digit of a(n) in base n is unequal to n-1, then the most significant digit of a(n) in base n equals 1.
%C From this it seems that, the least significant digit of a(n) in base n equals n-1 or the most significant digit of a(n) in base n equals 1, holds for all n > 1.
%C For n > 305 it seems that a(n) < n^2 - n - 1.
%C It seems that a(n) >= n*floor(3*n/4)-1; i.e. for any a(n) which is represented by a two-digit number in base n, the most significant digit is at least floor(3*n/4)-1. (End)
%C From _A.H.M. Smeets_, May 30 2019: (Start)
%C a(n) is a 5-digit number in base n representation for n in {2,3,4,5,7}.
%C a(n) is a 4-digit number in base n representation for n in {6,8,13}.
%C a(n) is a 3-digit number in base n representation for n in {9,10,11,12,14,15,16,17,18,21,25,34,35,52,71,72,73}.
%C For all other bases n, a(n) is a 2-digit number in base-n representation.
%C If a(n) = n*floor(3*n/4)-1, then n == 0 (mod 4) or n == 3 (mod 4). (End)
%H A.H.M. Smeets, <a href="/A060382/b060382.txt">Table of n, a(n) for n = 2..20000</a> (terms 2..3428 from Karl Hovekamp, with some corrections)
%H K. S. Brown, <a href="http://www.mathpages.com/home/kmath004/kmath004.htm">Digit Reversal Sums Leading to Palindromes</a>
%H A.H.M. Smeets, <a href="/A060382/a060382.txt">Lists of first 20 Lychrel numbers for bases n <= 1000</a>
%H A.H.M. Smeets, <a href="/A060382/a060382.gif">Scatterplot of log_10(a(n)/n^2) versus n for 73 < n <= 20000</a>
%e a(2) = 22 since A062129(k) > -1 (equivalently, A062131(k) > -1) for k < 22.
%o (Python)
%o def rev(n,base):
%o ....m = 0
%o ....while n > 0:
%o ........n, m = n//base, m*base+n%base
%o ....return m
%o n, a, steps = 2, 3, 0
%o while n <= 20000:
%o ....aa = a
%o ....ra = rev(a,n)
%o ....while aa != ra and steps < 1000:
%o ........aa = aa+ra
%o ........ra, steps = rev(aa,n), steps+1
%o ....if aa == ra:
%o ........a, aa, steps = a+1, a+1, 0
%o ....if steps == 1000:
%o ........print(n,a)
%o ........n, a, steps = n+1, n+2, 0 # _A.H.M. Smeets_, May 27 2019
%Y For the first palindrome in non-palindrome-free sequences, cf. A062129/A062131 (base 2), A033865 (base 10), A253241 (base 12).
%K nonn,base
%O 2,1
%A _Michel ten Voorde_, Apr 03 2001
%E More terms from _Karl Hovekamp_, Jan 03 2007