A060382 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.

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

Michel ten Voorde, Apr 03 2001

Keywords

• nonn
• base

Comments

Only a(2) is proved, all the others are conjectured. - Eric Chen, Apr 20 2015 [corrected by A.H.M. Smeets, May 27 2019]

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]

From A.H.M. Smeets, May 27 2019: (Start)

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.

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.

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.

For n > 305 it seems that a(n) < n^2 - n - 1.

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)

From A.H.M. Smeets, May 30 2019: (Start)

a(n) is a 5-digit number in base n representation for n in {2,3,4,5,7}.

a(n) is a 4-digit number in base n representation for n in {6,8,13}.

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}.

For all other bases n, a(n) is a 2-digit number in base-n representation.

If a(n) = n*floor(3*n/4)-1, then n == 0 (mod 4) or n == 3 (mod 4). (End)

Examples

			a(2) = 22 since A062129(k) > -1 (equivalently, A062131(k) > -1) for k < 22.
		

Links

• A.H.M. Smeets, Table of n, a(n) for n = 2..20000 (terms 2..3428 from Karl Hovekamp, with some corrections)
• K. S. Brown, Digit Reversal Sums Leading to Palindromes
• A.H.M. Smeets, Lists of first 20 Lychrel numbers for bases n <= 1000
• A.H.M. Smeets, Scatterplot of log_10(a(n)/n^2) versus n for 73 < n <= 20000

Crossrefs

For the first palindrome in non-palindrome-free sequences, cf. A062129/A062131 (base 2), A033865 (base 10), A253241 (base 12).

Programs

Python

def rev(n,base):
    m = 0
    while n > 0:
        n, m = n//base, m*base+n%base
    return m
n, a, steps = 2, 3, 0
while n <= 20000:
    aa = a
    ra = rev(a,n)
    while aa != ra and steps < 1000:
        aa = aa+ra
        ra, steps = rev(aa,n), steps+1
    if aa == ra:
        a, aa, steps = a+1, a+1, 0
    if steps == 1000:
        print(n,a)
        n, a, steps = n+1, n+2, 0 # A.H.M. Smeets, May 27 2019

Extensions

More terms from Karl Hovekamp, Jan 03 2007