A023109 -id:A023109 - cp's OEIS frontend

A281506 Numbers which require exactly 261 'Reverse and Add' steps to reach a palindrome.

1186060307891929990, 1186060317791929990, 1186060327691929990, 1186060337591929990, 1186060347491929990, 1186060357391929990, 1186060367291929990, 1186060377191929990, 1186060387091929990, 1186060407881929990, 1186060417781929990, 1186060427681929990, 1186060437581929990

Offset: 1

Andrey S. Shchebetov and Sergei D. Shchebetov, Jan 23 2017

The sequence starts with 1186060307891929990 (the 19-digit number also known as "the most delayed palindrome" and claimed as the world record, discovered by Jason Doucette on Nov 30 2005 and rediscovered by Vaughn Suite on Jan 02 2006) and continues for another 125 terms (none previously reported) each turning into a 119-digit palindrome after 261 steps until the sequence ends with 1186061987030929990. The distance between successive terms in the reported sequence has 9000000 as the greatest common divisor. No further numbers beyond 1186061987030929990 belonging to the same sequence are known, discovered or reported. The sequence was found empirically using computer modeling algorithms.
The sequence was extended to 108864 terms in total and ends with 1999291987030606810 - the last term of A281508 (see a-file). The sequence is complete - no further numbers beyond 1999291987030606810 belonging to the same sequence exist. The sequence was predicted theoretically and found empirically using computer modeling algorithms. - Sergei D. Shchebetov, May 12 2017
Comments from Sergei D. Shchebetov, Nov 14 2019: (Start)
There are two reasons that 1186060307891929990 cannot be the smallest term.
(1) Empirical: All numbers below were tested and none was found to have 261 (or higher) steps delay. This is presented, for example, in the Doucette link.
(2) Theoretical: There is no other combinations of the digits at 1186060307891929990 that gives you a lower number with the same reverse-and-sum result after the first step. This is because the number starts with 1 and you cannot go below 1 for the largest digit. Then it has 9s as the last 3 smallest digits and you cannot go up from there, but you could go down for the smallest digits (meaning up for the largest). For example, 1286060307891929980 (look at changes in the second digit from both ends: 1 turns into 2 and 9 turns into 8 with the sum staying 10 in both cases) would have the same 261-step delay.  Same is with 1386060307891929970, etc. If you calculate all possible combinations where the pairwise sum of the digits stays the same, you will get 108864 terms.
Also, since 2005, when 1186060307891929990 was discovered, people have checked all numbers up to 23-digit range and found none (except for our set) with 261-step (or higher) delays. So finding a number with a 288-step delay, as Rob van Nobelen did, was a real breakthrough.
(End)

			Each term requires exactly 261 steps to turn into a 119-digit palindrome, the last term of A281507, and is separated by some multiples of 9000000 from the adjacent sequence terms.

Popular Computing (Calabasas, CA), The 196 Problem, Vol. 3 (No. 30, Sep 1975).

Sergei D. Shchebetov, Table of n, a(n) for n = 1..126
Jason Doucette, World Records
Yutaka Nishiyama, Numerical Palindromes and the 196 Problem, International Journal of Pure and Applied Mathematics, Volume 80, No. 3, 2012, 375-384.
Sergei D. Shchebetov, 108864 terms (zipped file)
R. Styer, The Palindromic Conjecture and the Fibonacci Sequence, Villanova University, 1986, 1-11.
C. W. Trigg, Palindromes by Addition, Mathematics Magazine, 40 (1967), 26-28.
C. W. Trigg, More on Palindromes by Reversal-Addition, Mathematics Magazine, 45 (1972), 184-186.
Wikipedia, Lychrel Number
196 and Other Lychrel Numbers, 196 and Lychrel Number
Index entries for sequences related to Reverse and Add!

Cf. A023109, A033672, A065198, A065199, A065320, A065321, A065322, A065323, A065324, A065325, A065326, A065327, A070743, A072216, A072217, A072218, A281301, A281390, A281507.

A281507 Trajectory of 1186061987030929990 under the "Reverse and Add!" operation.

Original entry on oeis.org

1186061987030929990, 2185352294922536801, 3271704589845072613, 6434410079699144336, 12768830049399288682, 41457129443403175403, 71914259877895350817, 143719619755790592734, 581014717313707510075, 1151030424627424920260, 1771324671891665221771

Offset: 0

Andrey S. Shchebetov and Sergei D. Shchebetov, Jan 23 2017

1186061987030929990 is the largest of the first 126 numbers that require exactly 261 steps to turn into a palindrome (see A281506). The sequence reaches a 119-digit palindrome after 261 steps (see b-file). The number was obtained empirically using computer algorithms and was not reported before.

			a(1) = 1186061987030929990 + 999290307891606811 = 2185352294922536801.

Popular Computing (Calabasas, CA), The 196 Problem, Vol. 3 (No. 30, Sep 1975).

Sergei D. Shchebetov, Table of n, a(n) for n = 0..261
Jason Doucette, World Records
Yutaka Nishiyama, Numerical Palindromes and the 196 Problem, International Journal of Pure and Applied Mathematics, Volume 80 No. 3 2012, 375-384.
R. Styer, The Palindromic Conjecture and the Fibonacci Sequence, Villanova University, 1986, 1-11.
C. W. Trigg, Palindromes by Addition, Mathematics Magazine, 40 (1967), 26-28.
C. W. Trigg, More on Palindromes by Reversal-Addition, Mathematics Magazine, 45 (1972), 184-186.
Wikipedia, Lychrel Number
196 and Other Lychrel Numbers, 196 and Lychrel Number
Index entries for sequences related to Reverse and Add!

Cf. A023109, A033672, A065198, A065199, A065320, A065321, A065322, A065323, A065324, A065325, A065326, A065327, A070743, A072216, A072217, A072218, A281301, A281390, A281506.

k:=1186061987030929990; [n eq 1 select k else Self(n-1) + Seqint(Reverse(Intseq(Self(n-1)))): n in [1..20]]; // Bruno Berselli, Jan 23 2017

NestList[#+IntegerReverse[#]&,1186061987030929990,20] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 17 2019 *)

a(n+1) = a(n) + rev(a(n)).

A066058 In base 2: smallest integer which requires n 'Reverse and Add' steps to reach a palindrome.

Original entry on oeis.org

0, 2, 11, 44, 19, 20, 275, 326, 259, 202, 103, 74, 1027, 1070, 1049, 1072, 1547, 1310, 1117, 794, 569, 398, 3083, 2154, 1177, 1064, 4697, 4264, 4443, 2678, 2169, 1422, 779, 3226, 1551, 1114, 1815, 1062, 4197, 3106, 8697, 7238, 16633, 12302, 6683

Offset: 0

Klaus Brockhaus, Dec 04 2001

The analog of A023109 in base 2.

			11 is the smallest integer which requires two steps to reach a base 2 palindrome (cf. A066057), so a(2) = 11; written in base 10: 11 -> 11 + 13 = 24 -> 24 + 3 = 27; written in base 2: 1011 -> 1011 + 1101 = 11000 -> 11000 + 11 = 11011.

Cf. A066057, A023109, A062128, A062130, A033865, A006995, A057148.

(* For function b2reverse see A066057. *) function a066058(mx: integer); var k,m,n,rev,steps: integer; begin for k := 0 to mx do n := 0; steps := 0; m := n; rev := b2reverse(m); while not(steps = k and m = rev) do inc(n); m := n; rev := b2reverse(m); steps := 0; while steps < k and m <> rev do m := m + rev; rev := b2reverse(m); inc(steps); end; end; write(n,","); end; end; a066058(45);

Table[ SelectFirst[Range[0, 20000], (np = #; i = 0;
    While[ np != IntegerReverse[np, 2] && i <= n,
     np = np + IntegerReverse[np, 2]; i++];
i == n ) &] , {n, 0, 44}] (* Robert Price, Oct 16 2019 *)

def A066058(n):
    if n > 0:
        k = 0
        while True:
            m = k
            for i in range(n):
                s1 = format(m,'b')
                s2 = s1[::-1]
                if s1 == s2:
                    break
                m += int(s2,2)
            else:
                s1 = format(m,'b')
                if s1 == s1[::-1]:
                    return k
            k += 1
    else:
        return 0 # Chai Wah Wu, Jan 06 2015

A281508 Numbers requiring exactly 261 'Reverse and Add' steps to reach a palindrome.

Original entry on oeis.org

1999290307891606810, 1999290317791606810, 1999290327691606810, 1999290337591606810, 1999290347491606810, 1999290357391606810, 1999290367291606810, 1999290377191606810, 1999290387091606810, 1999290407881606810, 1999290417781606810, 1999290427681606810, 1999290437581606810

Offset: 1

Andrey S. Shchebetov and Sergei D. Shchebetov, Jan 24 2017

The sequence starts with 1999290307891606810 and continues for another 125 terms (none previously reported, including the first term) each turning into a 119-digit palindrome after 261 steps until the sequence ends with 1999291987030606810. The distance between successive terms in the reported sequence has 9000000 as the greatest common divisor. No further numbers beyond 1999291987030606810 belonging to the same sequence are known, discovered or reported. Moreover, 1999291987030606810 is currently the largest discovered "most delayed palindrome". The sequence was found empirically using computer modeling algorithms.

It is only a conjecture that there are no further terms. - N. J. A. Sloane, Jan 24 2017

			Each term requires exactly 261 steps to turn into a 119-digit palindrome, the last term of A281509, and is separated by some multiples of 9000000 from the adjacent sequence terms.

Popular Computing (Calabasas, CA), The 196 Problem, Vol. 3 (No. 30, Sep 1975).

Sergei D. Shchebetov, Table of n, a(n) for n = 1..126
Jason Doucette, World Records
Yutaka Nishiyama, Numerical Palindromes and the 196 Problem, International Journal of Pure and Applied Mathematics, Volume 80 No. 3 2012, 375-384.
R. Styer, The Palindromic Conjecture and the Fibonacci Sequence, Villanova University, 1986, 1-11.
C. W. Trigg, Palindromes by Addition, Mathematics Magazine, 40 (1967), 26-28.
C. W. Trigg, More on Palindromes by Reversal-Addition, Mathematics Magazine, 45 (1972), 184-186.
Wikipedia, Lychrel Number
196 and Other Lychrel Numbers, 196 and Lychrel Number
Index entries for sequences related to Reverse and Add!

Cf. A023109, A033672, A065198, A065199, A065320, A065321, A065322, A065323, A065324, A065325, A065326, A065327, A070743, A072216, A072217, A072218, A281301, A281390, A281506, A281507.

A281509 Trajectory of 1999291987030606810 (the largest presently known "most delayed palindrome") under the "Reverse and Add!" operation.

Original entry on oeis.org

1999291987030606810, 2185352294922536801, 3271704589845072613, 6434410079699144336, 12768830049399288682, 41457129443403175403, 71914259877895350817, 143719619755790592734, 581014717313707510075, 1151030424627424920260, 1771324671891665221771, 3542550333873429453542, 5996099577656760005995

Offset: 0

Andrey S. Shchebetov and Sergei D. Shchebetov, Jan 24 2017

1999291987030606810 is the largest of the 126 presently known numbers that require exactly 261 steps to turn into a palindrome (see A281508). It is also the largest discovered "most delayed palindrome". The sequence reaches a 119-digit palindrome after 261 steps (see b-file). The number was obtained empirically using computer algorithms and was not reported before.

			a(1) = 1999291987030606810 + 186060307891929991 = 2185352294922536801.

Popular Computing (Calabasas, CA), The 196 Problem, Vol. 3 (No. 30, Sep 1975).

Sergei D. Shchebetov, Table of n, a(n) for n = 0..261
Jason Doucette, World Records
Yutaka Nishiyama, Numerical Palindromes and the 196 Problem, International Journal of Pure and Applied Mathematics, Volume 80 No. 3 2012, 375-384.
R. Styer, The Palindromic Conjecture and the Fibonacci Sequence, Villanova University, 1986, 1-11.
C. W. Trigg, Palindromes by Addition, Mathematics Magazine, 40 (1967), 26-28.
C. W. Trigg, More on Palindromes by Reversal-Addition, Mathematics Magazine, 45 (1972), 184-186.
Wikipedia, Lychrel Number
196 and Other Lychrel Numbers, 196 and Lychrel Number
Index entries for sequences related to Reverse and Add!

Cf. A023109, A033672, A065198, A065199, A065320, A065321, A065322, A065323, A065324, A065325, A065326, A065327, A070743, A072216, A072217, A072218, A281301, A281390, A281506, A281507, A281508.

a(n+1) = a(n) + rev(a(n)).

a(n) = A281507(n) for n>0. - R. J. Mathar, Jan 27 2017

A326414 Numbers which require exactly 288 'Reverse and Add' steps to reach a palindrome.

Original entry on oeis.org

12000700000025339936491, 12000700001015339936491, 12000700002005339936491, 12000700010024339936491, 12000700011014339936491, 12000700012004339936491, 12000700020023339936491, 12000700021013339936491, 12000700022003339936491, 12000700030022339936491

Offset: 1

Andrey S. Shchebetov and Sergei D. Shchebetov, Oct 18 2019

Popular Computing (Calabasas, CA), The 196 Problem, Vol. 3 (No. 30, Sep 1975).

Sergei D. Shchebetov, Table of n, a(n) for n = 1..10000
Jason Doucette, World Records
Yutaka Nishiyama, Numerical Palindromes and the 196 Problem, International Journal of Pure and Applied Mathematics, Volume 80, No. 3, 2012, 375-384.
Sergei D. Shchebetov, 19353600 terms (zipped file)
R. Styer, The Palindromic Conjecture and the Fibonacci Sequence, Villanova University, 1986, 1-11.
C. W. Trigg, Palindromes by Addition, Mathematics Magazine, 40 (1967), 26-28.
C. W. Trigg, More on Palindromes by Reversal-Addition, Mathematics Magazine, 45 (1972), 184-186.
Wikipedia, Lychrel Number
196 and Other Lychrel Numbers, 196 and Lychrel Number
Index entries for sequences related to Reverse and Add!

Cf. A023109, A033672, A065198, A065199, A065320, A065321, A065322, A065323, A065324, A065325, A065326, A065327, A070743, A072216, A072217, A072218, A281301, A281390, A281506, A281507.

Each term requires exactly 288 steps to turn into a 142-digit palindrome.

Deleted an erroneous comment that said that the sequence was finite. - N. J. A. Sloane, Jun 23 2022

A015994 Smallest k such that the smallest palindrome > k in the Reverse and Add! trajectory of k is reached after exactly n iterations.

Original entry on oeis.org

1, 5, 59, 69, 166, 79, 188, 193, 1397, 829, 167, 2069, 1797, 849, 177, 999, 739, 1798, 989, 6999, 1297, 869, 187, 89, 10797, 10853, 10921, 10971, 13297, 10548, 13293, 17793, 20889, 700269, 106977, 108933, 80359, 13697, 10794, 15891, 1009227, 1007619, 1009246

Offset: 1

Felix Fröhlich, May 28 2022; original entry N. J. A. Sloane, Dec 11 1999

Variant of A023109 allowing k to be palindromic itself.

Smallest k such that A033665(k) = n.

Fabien Gelineau, Table of n, a(n) for n = 1..82

Cf. A023109.

iterationstosmallestpalindrome(n, bound) = my(x=n, i=0, d); while(1, if(i > bound, return(-1)); x=x+eval(concat(Vecrev(Str(x)))); i++; d=digits(x); if(d==Vecrev(d), return(i)))
a(n) = for(k=1, oo, if(iterationstosmallestpalindrome(k, n)==n, return(k))) \\ Felix Fröhlich, May 28 2022

Keyword "dead" removed, more terms added and entry revised by Felix Fröhlich, May 28 2022; Jun 22 2022

A077441 In base 4, smallest number that requires n Reverse and Add! steps to reach a palindrome.

Original entry on oeis.org

0, 4, 7, 26, 28, 127, 306, 348, 398, 301, 308, 203, 311, 783, 294, 350, 199, 296, 4268, 16595, 5326, 4253, 17399, 8235, 6189, 4270, 3107, 1270, 1532, 511, 67816, 65975, 24670, 12395, 4282, 3119, 28799, 16861, 18164, 66268, 45087, 71164, 309234

Offset: 0

Klaus Brockhaus, Nov 05 2002

Base-4 analog of A066058 (base 2) and A023109 (base 10).

			7 is the smallest number which requires two steps to reach a base 4 palindrome (cf. A075685), so a(2) = 5; 7 (decimal) = 13 -> 13 + 31 = 110 -> 110 + 011 = 121 (palindrome) = 25 (decimal).

Cf. A075685, A066058, A023109.

{m=46; v=[]; for(j=1,m+1,v=concat(v,-1)); mc=m+1; n=0; while(mc>0,a=-1; c=0; k=n; while(c0,d=divrem(q,4); q=d[1]; rev=4*rev+d[2]); if(k==rev,a=c; c=m+1,c++; k=k+rev)); if(0<=a&&a<=m,if(v[a+1]<0,v[a+1]=n; mc--; print1([a,n]))); n++); print(); for(j=1,m+1,print1(v[j],","))}

from gmpy2 import digits
def A077441(n):
    if n > 0:
        k = 0
        while True:
            m = k
            for i in range(n):
                s = digits(m,4)
                if s == s[::-1]:
                    break
                m += int(s[::-1],4)
            else:
                s = digits(m,4)
                if s == s[::-1]:
                    return k
            k += 1
    else:
        return 0 # Chai Wah Wu, Jan 17 2015

A090069 Numbers n such that there are (presumably) eight palindromes in the Reverse and Add! trajectory of n.

Original entry on oeis.org

3, 8, 20, 22, 100, 101, 116, 122, 139, 151, 160, 215, 221, 238, 313, 314, 320, 337, 343, 413, 436, 512, 611, 634, 696, 710, 717, 727, 733, 832, 931, 1004, 1011, 1070, 1101, 1160, 1250, 1340, 1430, 1520, 1610, 1700, 1771, 2000, 2002, 2003, 2010, 2100, 2112

Offset: 1

Klaus Brockhaus, Nov 20 2003

For terms <= 5000 each palindrome is reached from the preceding one or from the start in at most 15 steps; after the presumably last one no further palindrome is reached in 2000 steps.

			The trajectory of 8 begins 8, 16, 77, 154, 605, 1111, 2222, 4444, 8888, 17776, 85547, 160105, 661166, 1322332, 3654563, 7309126, ...; at 7309126 it joins the (presumably) palindrome-free trajectory of A063048(7) = 10577, hence 8, 77, 1111, 2222, 4444, 8888, 661166 and 3654563 are the eight palindromes in the trajectory of 8 and 8 is a term.

Index entries for sequences related to Reverse and Add!

Cf. A023108, A023109, A065001, A070742, A077594.

A090070 Numbers n such that there are (presumably) nine palindromes in the Reverse and Add! trajectory of n.

Original entry on oeis.org

4, 10, 11, 535, 1000, 1001, 10007, 10101, 20006, 30005, 50003, 60002, 70001, 80000, 80008, 100070, 110060, 120050, 130040, 140030, 150020, 160010, 170000, 170071, 200000, 200002, 1000003, 1000150, 1001001, 1010050, 1100140, 1110040, 1200130

Offset: 1

Klaus Brockhaus, Nov 20 2003

For terms < 5000000 each palindrome is reached from the preceding one or from the start in at most 35 steps; after the presumably last one no further palindrome is reached in 2000 steps.

			The trajectory of 4 begins 4, 8, 16, 77, 154, 605, 1111, 2222, 4444, 8888, 17776, 85547, 160105, 661166, 1322332, 3654563, 7309126, ...; at 7309126 it joins the (presumably) palindrome-free trajectory of A063048(7) = 10577, hence 4, 8, 77, 1111, 2222, 4444, 8888, 661166 and 3654563 are the nine palindromes in the trajectory of 4 and 4 is a term.

Index entries for sequences related to Reverse and Add!

Cf. A023108, A023109, A065001, A070742, A077594.

cp's OEIS Frontend

A281506 Numbers which require exactly 261 'Reverse and Add' steps to reach a palindrome.

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A281507 Trajectory of 1186061987030929990 under the "Reverse and Add!" operation.

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A066058 In base 2: smallest integer which requires n 'Reverse and Add' steps to reach a palindrome.

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A281508 Numbers requiring exactly 261 'Reverse and Add' steps to reach a palindrome.

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A281509 Trajectory of 1999291987030606810 (the largest presently known "most delayed palindrome") under the "Reverse and Add!" operation.

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A326414 Numbers which require exactly 288 'Reverse and Add' steps to reach a palindrome.

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A015994 Smallest k such that the smallest palindrome > k in the Reverse and Add! trajectory of k is reached after exactly n iterations.

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PARI

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A077441 In base 4, smallest number that requires n Reverse and Add! steps to reach a palindrome.

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