A077404 -id:A077404 - cp's OEIS frontend

A077405 Trajectory of n under the Reverse and Add! operation carried out in base 3 (presumably) does not reach a palindrome and (presumably) does not join the trajectory of any term m < n.

103, 746, 805, 2231, 2326, 2671, 2725, 2959, 2969, 3679, 4421, 4430, 4439, 4448, 5894, 6626, 6638, 6686, 6698, 6733, 6741, 6779, 6789, 6793, 6943, 7124, 7365, 7849, 8093, 8801, 8836, 10771, 11078, 11158, 13184, 13361, 17558, 17639, 19115, 19196, 19733, 19895, 19901, 19907, 20106, 20124, 20149, 20161

Offset: 1

Klaus Brockhaus, Nov 05 2002

Base 3 analog of A075252 (base 2), A075421 (base 4) and A063048 (base 10); subsequence of A077404. - A proof that the base 3 trajectory does not contain a palindrome has been found up to now for none of the terms. - If the trajectory of an integer k joins the trajectory of a smaller integer which is a term of the present sequence, then this occurs after very few Reverse and Add! steps (at most 9 for k < 20000). On the other hand, the trajectories of the terms of this sequence do not join the trajectory of any smaller term within at least 1000 steps.

			805 is a term since the trajectory of 805 (presumably) does not lead to a number which occurs in the trajectory of 103 or of 746.

A.H.M. Smeets, Table of n, a(n) for n = 1..529
Index entries for sequences related to Reverse and Add!

Cf. A063048, A075252, A075421, A077404.

limit = 10^3; (* Assumes that there is no palindrome if none is found before "limit" iterations *)
utraj = {};
Select[Range[0,21000], (i = 0;
   x = NestWhileList[(i++; # + IntegerReverse[#, 3]) &, #, # !=
        IntegerReverse[#, 3] && i < limit  &];
   If[i >= limit  && Intersection[x, utraj] == {},
    utraj = Union[utraj, x]; True,
    utraj = Union[utraj, x]]) &]
(* Robert Price, Oct 19 2019 *)

Offset changed to 1 by A.H.M. Smeets, Feb 14 2019

a(41)-a(48) from A.H.M. Smeets, Feb 18 2019

A348570 Positive integers which apparently never result in a palindrome under repeated applications of the function f(x) = x + (x with digits in Zeckendorf representation reversed). Zeckendorf representation analog of Lychrel numbers.

Original entry on oeis.org

59, 61, 69, 75, 77, 100, 105, 113, 115, 122, 128, 130, 131, 135, 136, 140, 142, 143, 148, 151, 153, 160, 162, 163, 166, 172, 177, 180, 183, 188, 191, 192, 196, 198, 200, 209, 210, 212, 215, 222, 223, 229, 230, 231, 237, 240, 249, 250, 257, 258, 263, 264, 266

Offset: 1

A.H.M. Smeets, Oct 23 2021

Zeckendorf representation version of A023108 (base 10).
For the Zeckendorf representation of numbers see A014417.
For palindromic numbers in Zeckendorf representation see A094202.
The "Reverse and Add!" operation (A349239) applied in Zeckendorf representation seems to behave similarly to the "Reverse and Add!" operation applied in any fixed-base representation. The first 53 terms are however obtained after performing 10^4 "Reverse and Add!" steps (see Python program).
For records and record-setting values in the number of "Reverse and Add!" steps see A348572 and A348571 respectively.
Do any of these numbers have a trajectory in which the Lychrel property can be proved (like 22 in base 2 as in A061561)?
Iteration steps are given by n := n+A349238(n), or n := A349239(n).
Closure of reverse operation is given by: Let Z be the regular expression for numbers in Zeckendorf representation, Z = 0|(100*)*10*, and L(Z) its corresponding regular language. Then for s in L(Z), the reversal of s is in L(0*)L(Z).
Let h be the homomorphism from Zeckendorf representation to a conventional radix representation, then addition in Zeckendorf representation, +_Z, is given by z1 +_Z z2 = h^(-1)(h(z1) + h(z2)). A direct method for addition in Zeckendorf representation is given by Ahlbach et al.

A.H.M. Smeets, Table of n, a(n) for n = 1..20000
C. Ahlbach, J. Usatine, C. Frougny, and N. Pippenger, Efficient algorithms for Zeckendorf arithmetic, Fibonacci Quart. 51, no. 3 (2013), 249-255.

Cf. A014417, A061561, A094202, A348571, A348572, A349238, A349239.

Lychrel numbers in fixed bases: A066059 (base 2), A077404 (base 3), A075420 (base 4), A023108 (base 10).

# Using functions NumToFib and RevFibToNum from A349238.
n, a = 0, 0
while n < 53:
    a += 1
    aa, sa = a, NumToFib(a)
    ar, s = RevFibToNum(sa), 0
    while aa != ar and s < 10000:
        s, aa = s+1, aa+ar
        sa = NumToFib(aa)
        ar = RevFibToNum(sa)
    if aa != ar:
        n += 1
        print(a, end = ", ")

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.

Original entry on oeis.org

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

Michel ten Voorde, Apr 03 2001

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)

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

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

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

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

More terms from Karl Hovekamp, Jan 03 2007

A077402 Reverse and Add! carried out in base 3; number of steps needed to reach a palindrome, or -1 if no palindrome is ever reached.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 1, 2, 0, 1, 0, 2, 1, 0, 2, 3, 0, 4, 1, 2, 0, 1, 2, 0, 3, 4, 0, 1, 0, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 0, 2, 3, 3, 2, 1, 1, 2, 3, 3, 2, 3, 0, 18, 1, 2, 0, 1, 2, 4, 1, 2, 2, 1, 2, 4, 1, 2, 0, 3, 2, 4, 1, 2, 2, 3, 2, 4, 17, 18, 0, 1, 0, 2, 1, 1, 2, 1, 1, 3, 1, 0, 2, 1, 1, 16, 1, 1, 2, 2, 0, 2, 4, -1, 16, 3, 15, 2, 1, 1, 2, 1, 0, 3, 3, 3, 2, 1, 1, 16, 1

Offset: 0

Klaus Brockhaus, Nov 05 2002

Base-3 analog of A066057 (base 2), A075685 (base 4) and A033665 (base 10). a(103) = -1 is a conjecture (cf. A066450, A077408). For values of n such that presumably a(n) = -1 see A077404.

			17 (decimal) = 122 -> 122 + 221 = 1120 -> 1120 + 211 = 2101 -> 2101 + 1012 = 10120 -> 10120 + 2101 = 12221 (palindrome) = 160 (decimal) requires 4 steps, so a(17) = 4.

Index entries for sequences related to Reverse and Add!

Cf. A014190, A066450, A033665, A066057, A075685, A077404, A077405.

m := 120; stop := 1000; for n := 0 to m do v := -1; c := 0; k := n; while c < stop do d := k; rev := 0; while d > 0 do rev := 3*rev + (d mod 3); d := d div 3; end; if k = rev then v := c; c := stop; else inc(c); k := k + rev; end; end; write(v,","); end;

A077406 In base 3: n sets a new record for the number of Reverse and Add! steps needed to reach a palindrome starting with n.

Original entry on oeis.org

0, 3, 5, 15, 17, 53, 1466, 9717, 32857, 37340, 37366, 37367, 531773, 1596329

Offset: 1

Klaus Brockhaus, Nov 05 2002

RECORDS transform of A077402. - Base-3 analog of A066144 (base 2), A075686 (base 4) and A065198 (base 10). Integers like 103, for which a palindrome is (presumably) never reached (cf. A077404), are of course disregarded. A077407 gives the corresponding records.

			Starting with 15, 3 Reverse and Add! steps are needed to reach a palindrome; starting with n < 15, less (at most 2) steps are needed.

N. J. A. Sloane, Transforms
Index entries for sequences related to Reverse and Add!

Cf. A065198, A066144, A075686, A077402, A077404, A077407.

a(14) from Sean A. Irvine, Apr 19 2010

A306365 Trajectory of n under the Reverse and Add! operation carried out in base 5 (presumably) does not reach a palindrome.

Original entry on oeis.org

708, 718, 723, 731, 733, 743, 828, 838, 843, 851, 853, 863, 958, 963, 983, 1078, 1083, 1103, 1203, 1299, 1309, 1332, 1342, 1347, 1350, 1355, 1357, 1359, 1367, 1419, 1429, 1452, 1462, 1467, 1475, 1477, 1479, 1487, 1499, 1539, 1582, 1607, 1619, 1659, 1702, 1707, 1727, 1739, 1779, 1827, 1859, 1923, 1933, 1956

Offset: 1

A.H.M. Smeets, Feb 10 2019

Base-5 analog of A066059 (base 2), A077404 (base 3), A075420 (base 4) and A023108 (base 10).

All terms are tested up to 200 iteration steps, i.e., within 200 steps no palindrome was reached.

A.H.M. Smeets, Table of n, a(n) for n = 1..20000

Cf. A066059, A075420, A077404, A023108.

cp's OEIS Frontend

A077405 Trajectory of n under the Reverse and Add! operation carried out in base 3 (presumably) does not reach a palindrome and (presumably) does not join the trajectory of any term m < n.

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A348570 Positive integers which apparently never result in a palindrome under repeated applications of the function f(x) = x + (x with digits in Zeckendorf representation reversed). Zeckendorf representation analog of Lychrel numbers.

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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.

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A077402 Reverse and Add! carried out in base 3; number of steps needed to reach a palindrome, or -1 if no palindrome is ever reached.

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A077406 In base 3: n sets a new record for the number of Reverse and Add! steps needed to reach a palindrome starting with n.

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A306365 Trajectory of n under the Reverse and Add! operation carried out in base 5 (presumably) does not reach a palindrome.

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