A023108 -id:A023108 - cp's OEIS frontend

A063060 'Reverse and Add!' trajectory of 10553.

10553, 46054, 91118, 172237, 904508, 1709917, 8908988, 17807086, 85877957, 161855815, 680413976, 1359728062, 3968007593, 7925016286, 14751121583, 53263237324, 95636473559, 191173937218, 1003913308409, 10051946501410

Offset: 0

Klaus Brockhaus, Jul 07 2001

			a(1) = 10553 + 35501 = 46054.

Reinhard Zumkeller and Michael Lee, Table of n, a(n) for n = 0..2423 (terms 0 through 250 provided by Reinhard Zumkeller)
Index entries for sequences related to Reverse and Add!

A033865, A023108, A006960, A063061.

Cf. A056964, A004086.

m := 10553; stop := 25; c := 0; rev := int_reverse(m); while m <> rev and c < stop do inc(c); write(m," "); m := m + rev; rev := int_reverse(m); end;

a063060 n = a063060_list !! n
a063060_list = iterate a056964 10553 -- Reinhard Zumkeller, Sep 22 2011

NestList[# + FromDigits[Reverse[IntegerDigits[#]]]&, 10553, 40] (* Vincenzo Librandi, May 03 2014 *)

Updated b-file from Michael Lee, Apr 01 2012

A063063 'Reverse and Add!' trajectory of 10563.

Original entry on oeis.org

10563, 47064, 93138, 176277, 948948, 1798797, 9777768, 18455547, 93011028, 175022067, 935242638, 1771485177, 9487326948, 17983564797, 97730103768, 184460207547, 930162272028, 1750434533067, 9353788873638, 17717577747177

Offset: 0

Klaus Brockhaus, Jul 07 2001

			a(1) = 10563 + 36501 = 47064.

Reinhard Zumkeller and Michael Lee, Table of n, a(n) for n = 0..2383 (terms 0 through 250 provided by Reinhard Zumkeller)
Index entries for sequences related to Reverse and Add!

A033865, A023108, A006960, A063064.

Cf. A056964, A004086.

m := 10563; stop := 25; c := 0; rev := int_reverse(m); while m <> rev and c < stop do inc(c); write(m," "); m := m + rev; rev := int_reverse(m); end;

a063063 n = a063063_list !! n
a063063_list = iterate a056964 10563 -- Reinhard Zumkeller, Sep 22 2011

NestList[# + FromDigits[Reverse[IntegerDigits[#]]]&, 10563, 30] (* Vincenzo Librandi, May 03 2014 *)

A066054 'Reverse and Add!' trajectory of 10583.

Original entry on oeis.org

10583, 49084, 97178, 184357, 937838, 1776577, 9533348, 17966707, 88733678, 176367466, 841131137, 1572262285, 7394885036, 13700769973, 51697470704, 92404950319, 183710890748, 1030808908129, 10248906988430, 13737867972631

Offset: 0

Klaus Brockhaus, Nov 30 2001

			a(1) = 10583 + 38501 = 49084.

Harry J. Smith and Vincenzo Librandi, Table of n, a(n) for n = 0..1000 (the first 150 terms from Harry J. Smith)
Index entries for sequences related to Reverse and Add!

Cf. A033865, A023108, A006960, A066055.

Cf. A056964, A004086.

: m := 10583; stop := 20; c := 0; rev := int_reverse(m); while m <> rev and c < stop do inc(c); write(m," "); m := m + rev; rev := int_reverse(m); end;

a066054 n = a066054_list !! n
a066054_list = iterate a056964 10583 -- Reinhard Zumkeller, Sep 22 2011

NestList[# + FromDigits[Reverse[IntegerDigits[#]]]&,  10583, 40] (* Vincenzo Librandi, May 03 2014 *)

Rev(x)= { local(d, r=0); while (x>0, d=x%10; x\=10; r=r*10 + d); return(r) } { a=10583; for (n = 0, 150, if (n, a+=Rev(a)); write("b066054.txt", n, " ", a) ) } \\ Harry J. Smith, Nov 08 2009

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 = ", ")

A030547 Number of terms (including the initial term) needed to reach a palindrome when the Reverse Then Add! map (x -> x + (x-with-digits-reversed)) is repeatedly applied to n, or -1 if a palindrome is never reached.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 1, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 1, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 1, 2, 3, 2, 3, 3, 2, 2, 2, 2, 2, 1, 2, 3, 3, 4, 2, 2, 2, 2, 3, 2, 1, 3, 4, 5, 2, 2, 2, 3, 2, 3, 3, 1, 5, 7, 2, 2, 3, 2, 3, 3, 4, 5, 1, 25, 2, 3, 2, 3, 3, 4, 5, 7, 25

Offset: 1

Eric W. Weisstein

It is conjectured that a(196) is the smallest term equal to -1. See A023108.

Daniel Lignon, Dictionnaire de (presque) tous les nombres entiers, Ellipses, Paris, 2012, 702 pages. See Entry 196.

Eric Weisstein's World of Mathematics, 196 Algorithm.

Cf. A006960, A023108, A063018, etc.

Equals A033665(n) + 1.

Table[Length@
  NestWhileList[# + IntegerReverse[#] &, n, ! PalindromeQ[#]  &], {n, 98}] (* Robert Price, Oct 18 2019 *)

Edited by N. J. A. Sloane, May 09 2015

A063052 Integers n > 879 such that the 'Reverse and Add!' trajectory of n joins the trajectory of 879.

Original entry on oeis.org

978, 1497, 1587, 1677, 1767, 1857, 1947, 2496, 2586, 2676, 2766, 2856, 2946, 3495, 3585, 3675, 3765, 3855, 3945, 4494, 4584, 4674, 4764, 4854, 4944, 5493, 5583, 5673, 5763, 5853, 5943, 6492, 6582, 6672, 6762, 6852, 6942, 7491, 7581, 7671, 7761, 7851

Offset: 1

Klaus Brockhaus, Jul 07 2001

Subsequence of A023108.

			The trajectory of 1497 reaches 9438 in one step and 9438 is a term in the trajectory of 879, so 1497 belongs to the present sequence. The corresponding term in A063053, giving the number of steps, accordingly is 1.

Index entries for sequences related to Reverse and Add!

Cf. A033865, A023108, A063051, A063053.

Offset corrected by Sean A. Irvine, Apr 17 2023

A063055 Integers n > 1997 such that the 'Reverse and Add!' trajectory of n joins the trajectory of 1997.

Original entry on oeis.org

2996, 3995, 5993, 6992, 7991, 8089, 8179, 8269, 8359, 8449, 8539, 8629, 8719, 8809, 8899, 8989, 8990, 9088, 9178, 9268, 9358, 9448, 9538, 9628, 9718, 9808, 9898, 9988, 13397, 14387, 15377, 16367, 17357, 17897, 18347, 18887, 19337, 19877, 23396

Offset: 0

Klaus Brockhaus, Jul 07 2001

Subsequence of A023108.

			The trajectory of 3995 reaches 9988 in one step and 9988 is a term in the trajectory of 1997, so 3995 belongs to the present sequence. The corresponding term in A063056, giving the number of steps, accordingly is 1.

Index entries for sequences related to Reverse and Add!

A033865, A023108, A063054, A063056.

A063058 Integers n > 7059 such that the 'Reverse and Add!' trajectory of n joins the trajectory of 7059.

Original entry on oeis.org

7149, 7239, 7329, 7419, 7509, 7599, 7689, 7779, 7869, 7959, 8058, 8148, 8238, 8328, 8418, 8508, 8598, 8688, 8868, 8958, 9057, 9147, 9237, 9327, 9417, 9507, 9597, 9687, 9777, 9867, 9957, 13596, 14586, 15576, 16566, 17292, 17556, 18096, 18282, 18546

Offset: 0

Klaus Brockhaus, Jul 07 2001

Subsequence of A023108.

			The trajectory of 7239 reaches 16566 in one step and 16566 is a term in the trajectory of 7059, so 7239 belongs to the present sequence. The corresponding term in A063059, giving the number of steps, accordingly is 1.

Index entries for sequences related to Reverse and Add!

A033865, A023108, A063057, A063059.

A063061 Integers n > 10553 such that the 'Reverse and Add!' trajectory of n joins the trajectory of 10553.

Original entry on oeis.org

11543, 12097, 12533, 13087, 13523, 14077, 14513, 15067, 15503, 16057, 16597, 17047, 17587, 18037, 18577, 19027, 19567, 20552, 21542, 22096, 22532, 23086, 23522, 24076, 24512, 25066, 25502, 26056, 26596, 27046, 27586, 28036, 28576

Offset: 0

Klaus Brockhaus, Jul 07 2001

Subsequence of A023108.

			The trajectory of 12097 reaches 91118 in one step and 91118 is a term in the trajectory of 10553, so 12097 belongs to the present sequence. The corresponding term in A063062, giving the number of steps, accordingly is 1.

Index entries for sequences related to Reverse and Add!

A033865, A023108, A063060, A063062.

A063064 Integers n > 10563 such that the 'Reverse and Add!' trajectory of n joins the trajectory of 10563.

Original entry on oeis.org

11553, 12543, 13533, 14097, 14523, 15087, 15513, 16077, 16503, 17067, 18057, 18597, 19047, 19587, 20562, 21552, 22542, 24096, 24522, 25086, 25512, 26076, 26502, 27066, 28056, 28596, 29046, 29586, 30561, 31551, 32541, 33531, 34095

Offset: 0

Klaus Brockhaus, Jul 07 2001

Subsequence of A023108.

			The trajectory of 12543 reaches 47064 in one step and 47064 is a term in the trajectory of 10563, so 12543 belongs to the present sequence. The corresponding term in A063065, giving the number of steps, accordingly is 1.

Index entries for sequences related to Reverse and Add!

A033865, A023108, A063063, A063065.

cp's OEIS Frontend

A063060 'Reverse and Add!' trajectory of 10553.

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A063063 'Reverse and Add!' trajectory of 10563.

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A066054 'Reverse and Add!' trajectory of 10583.

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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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A030547 Number of terms (including the initial term) needed to reach a palindrome when the Reverse Then Add! map (x -> x + (x-with-digits-reversed)) is repeatedly applied to n, or -1 if a palindrome is never reached.

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A063052 Integers n > 879 such that the 'Reverse and Add!' trajectory of n joins the trajectory of 879.

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A063055 Integers n > 1997 such that the 'Reverse and Add!' trajectory of n joins the trajectory of 1997.

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A063058 Integers n > 7059 such that the 'Reverse and Add!' trajectory of n joins the trajectory of 7059.

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