A033865 -id:A033865 - cp's OEIS frontend

A063057 'Reverse and Add!' trajectory of 7059.

7059, 16566, 83127, 155265, 717816, 1336533, 4692864, 9375828, 17661567, 94178238, 177465387, 961030158, 1812060327, 9042662508, 17095324917, 89037683988, 177976357086, 858730036857, 1617360074715, 6792060711876

Offset: 0

Klaus Brockhaus, Jul 07 2001

			a(1) = 7059 + 9507 = 16566.

Reinhard Zumkeller and Michael Lee, Table of n, a(n) for n = 0..2432 (terms 0 through 250 provided by Reinhard Zumkeller)
Index entries for sequences related to Reverse and Add!

Cf. A033865, A023108, A006960, A063058, A056964, A004086.

m := 7059; 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;

a063057 n = a063057_list !! n
a063057_list = iterate a056964 7059 -- Reinhard Zumkeller, Sep 22 2011

NestList[# + FromDigits[Reverse[IntegerDigits[#]]]&, 7059, 40] (* Vincenzo Librandi, May 03 2014 *)

A063054 'Reverse and Add!' trajectory of 1997.

Original entry on oeis.org

1997, 9988, 18887, 97768, 184547, 930028, 1750067, 9350638, 17711177, 94822948, 179745797, 977293768, 1844686547, 9301551028, 17503102067, 93523232638, 177146465177, 948711106948, 1798312224797, 9772534363768, 18446168716547

Offset: 0

Klaus Brockhaus, Jul 07 2001

			a(1) = 1997 + 7991 = 9988.

Reinhard Zumkeller and Michael Lee, Table of n, a(n) for n = 0..2318 (terms 0 through 250 provided by Reinhard Zumkeller)
Index entries for sequences related to Reverse and Add!

A033865, A023108, A006960, A063055, A056964, A004086.

m := 1997; 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;

a063054 n = a063054_list !! n
a063054_list = iterate a056964 1997 -- Reinhard Zumkeller, Sep 22 2011

NestList[ #+FromDigits[ Reverse[ IntegerDigits[ # ] ] ]&, 1997, 25 ]
NestList[#+IntegerReverse[#]&,1997,25] (* Harvey P. Dale, Jul 15 2025 *)

A063060 'Reverse and Add!' trajectory of 10553.

Original entry on oeis.org

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

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

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.

cp's OEIS Frontend

A063057 'Reverse and Add!' trajectory of 7059.

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A063054 'Reverse and Add!' trajectory of 1997.

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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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PARI

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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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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