A348572 -id:A348572 - cp's OEIS frontend

A066145 In base 2, records for the number of 'Reverse and Add' steps needed to reach a palindrome.

0, 1, 2, 4, 5, 11, 21, 32, 37, 46, 48, 49, 53, 89, 99, 142, 147, 273, 297, 345, 515, 550, 573

Offset: 1

Klaus Brockhaus, Dec 08 2001

The analog of A065199 in base 2. A066144 gives the corresponding starting points.

Terms a(19..22) obtained by assuming that a(n+1) <= a(n) + 300. - A.H.M. Smeets, Apr 30 2022

			Starting with 74, 11 'Reverse and Add' steps are needed to reach a palindrome; starting with n < 74, at most 5 steps are needed.

Index entries for sequences related to Reverse and Add!

Record values in base b: A077407 (b=3), A075687 (b=4), A306600 (b=8), A065199 (b=10), A348572 (Zeckendorf).

Cf. A062130, A066057, A066058, A066059, A066144.

limit = 10^3; (* Assumes that there is no palindrome if none is found before "limit" iterations *)
best = -1; lst = {};
For[n = 0, n <= 10000, n++,
np = n; i = 0;
While[np != IntegerReverse[np, 2] && i < limit,
  np = np + IntegerReverse[np, 2]; i++];
If[i < limit && i > best, best = i; AppendTo[lst, i]]]; lst (* Robert Price, Oct 14 2019 *)

Offset corrected and a(19)-a(23) by A.H.M. Smeets, Apr 30 2022

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 = ", ")

A348571 In Zeckendorf representation: integers that set a new record for the number of Reverse and Add steps (A349239) needed to reach a palindrome (A094202).

Original entry on oeis.org

0, 2, 7, 20, 54, 63, 114, 1002, 1413, 3007, 4447, 35131, 599185, 2189416, 2738842, 3253273, 108250112

Offset: 1

A.H.M. Smeets, Oct 23 2021

Corresponding record values in A348572.
For Zeckendorf representation of numbers see A014417.
Lychrel numbers, as given in A348570, are excluded from this list because it is believed that those numbers never reach a palindrome.

			Trajectory of 20, i.e., 101010 in Zeckendorf representation:
       101010 + 010101      =      1010100
      1010100 + 0010101     =     10010010
     10010010 + 01001001    =    100100100
    100100100 + 001001001   =   1000010001
   1000010001 + 1000100001  =  10100000010
  10100000010 + 01000000101 = 100100001001, which is palindromic.
Due to the fact that any number smaller than 20 reaches a palindrome in fewer than 6 steps, 20 is a record-setting nonnegative integer.
The Lychrel numbers, as given in A348570, are excluded, because it is believed that those numbers never reach a palindromic number.

A.H.M. Smeets, Python program

Cf. A014417 (Zeckendorf digits), A349239 (reverse and add), A094202 (palindromes).

Cf. A348572 (number of steps), A348570 (Lychrels).

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A066145 In base 2, records for the number of 'Reverse and Add' steps needed to reach a palindrome.

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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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A348571 In Zeckendorf representation: integers that set a new record for the number of Reverse and Add steps (A349239) needed to reach a palindrome (A094202).

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