A006960 -id:A006960 - cp's OEIS frontend

A331556 The lower (or left) offset of a 196-iterate (A006960) from the largest palindrome less than the iterate.

5, 9, 14, 99, 52, 89, 100, 407, 268, 10769, 10890, 99, 99, 4400, 8900, 9890, 10000, 97625, 1089, 3584, 99, 629882, 1099890, 10989, 926, 890000, 8491505, 10890099, 8229644, 9999989, 69923062, 10890000, 99099000, 43337905, 99990089, 962943454, 109890, 454649691

Offset: 1

James D. Klein, Jan 20 2020

When normalized over (0,1) by their respective palindrome-free interval about a 196-iterate, it has been empirically observed that the frequency distribution of this sequence appears to be quite symmetric about 0.5, as well as fractal when plotting the distribution over decreasing bin sizes.

The 196-iterates referred to here come from the reverse-and-add process generating A006960.

			The first term is 5 since 196-191 = 5
The second term is 9 since 887-878 = 9, etc.

Cf. A006960, A331557, A331560.

Map[Block[{k = # - 1}, While[k != IntegerReverse@ k, k--]; # - k] &, NestList[# + IntegerReverse[#] &, 196, 25]] (* brute force, or *)
Map[# - Block[{n = #, w, len, ww}, w = IntegerDigits[n]; len = Length@ w; ww = Take[w, Ceiling[len/2] ]; If[# < n, #, FromDigits@ Flatten@{#, If[OddQ@ len, Reverse@ Most@ #, Reverse@ #]} &@ If[Last@ ww == 0, MapAt[# - 1 &, Most@ ww, -1]~Join~{9}, MapAt[# - 1 &, ww, -1]]] &@ FromDigits@ Flatten@ {ww, If[OddQ@ len, Reverse@ Most@ ww, Reverse@ ww]}] &, NestList[# + IntegerReverse[#] &, 196, 37]] (* Michael De Vlieger, Jan 22 2020 *)

# Slow Brute-force
n = 196
while n < 10**15:
  m = n
  while m != int(str(m)[::-1]): m+=-1
  print(n-m, end=', ')
  n = n + int(str(n)[::-1])

a(n) = A331560(n) - A331557(n).

More terms from Michael De Vlieger, Jan 22 2020

A331557 The upper (or right) offset of a 196-iterate (A006960) from the smallest palindrome greater than the iterate.

Original entry on oeis.org

6, 1, 96, 11, 48, 11, 10, 693, 732, 231, 110, 10901, 10901, 5600, 1100, 110, 1000, 12375, 108911, 96416, 99901, 470118, 110, 1089011, 999074, 110000, 2508495, 109901, 1770356, 11, 40076938, 99110000, 10901000, 56662095, 9911, 137056546, 1099890110, 545350309

Offset: 1

James D. Klein, Jan 20 2020

When normalized over (0,1) by their respective palindrome-free interval about a 196-iterate, it has been empirically observed that the frequency distribution of this sequence appears to be quite symmetric about 0.5, as well as fractal when plotting the distribution over decreasing bin sizes.

The 196-iterates referred to here come from the reverse-and-add process generating A006960.

			The first term is 6 since 202-196 = 6;
The second term is 1 since 888-887 = 1; etc.

Cf. A006960, A331556, A006960.

# Upper 196 offsets. Slow brute force
n = 196
while n < 10**15:
  m = n
  while m != int(str(m)[::-1]): m+=1
  print(m-n)
  n = n + int(str(n)[::-1])

a(n) = A331560(n) - A331556(n).

More terms from Jinyuan Wang, Feb 29 2020

A331560 Size of the palindrome-free intervals about the 196-iterates, A006960.

Original entry on oeis.org

11, 10, 110, 110, 100, 100, 110, 1100, 1000, 11000, 11000, 11000, 11000, 10000, 10000, 10000, 11000, 110000, 110000, 100000, 100000, 1100000, 1100000, 1100000, 1000000, 1000000, 11000000, 11000000, 10000000, 10000000, 110000000, 110000000, 110000000, 100000000

Offset: 1

James D. Klein, Jan 20 2020

By empirical observation, the integers in this sequence are of the form 10*10^n and 11*10^n, n >= 0, since they are the difference of consecutive palindromes surrounding the 196-iterates. (No differences of 2 observed.)

			191 < 196 < 202, 202 - 191 = 11;
878 < 887 < 888, 888 - 878 = 10; etc.

Cf. A006960, A331556, A331557.

# Palindrome-free interval about 196 offsets. Slow brute-force
n = 196
while n < 10**15:
     m = n
     while m != int(str(m)[::-1]): m+=1
     s = m
     m = n
     while m != int(str(m)[::-1]): m-=1
     print(s-m)
     n = n + int(str(n)[::-1])

a(n) = A331556(n) + A331557(n).

a(31)-a(34) from Jinyuan Wang, Feb 29 2020

A217774 Palindromic deviation of terms of A006960.

Original entry on oeis.org

5, 1, 5, 2, 7, 2, 1, 14, 16, 11, 2, 2, 2, 8, 2, 2, 1, 14, 4, 14, 2, 28, 11, 20, 15, 2, 21, 13, 19, 2, 17, 11, 11, 27, 4, 32, 4, 37, 12, 12, 33, 2, 13, 28, 5, 49, 17, 47, 13, 43, 6, 39, 17, 44, 6, 45, 2, 35, 8, 37, 6, 69, 15, 47, 18, 48, 10, 33, 2, 59, 15, 19, 26, 17, 17, 73, 15, 55, 15, 63, 15, 43, 17, 12, 67, 10

Offset: 0

W. W. Kokko, Mar 24 2013

nonn

Apply the map n -> PD(n) = A064834(n) to the terms of A006960.

If 196 ever reaches a palindrome (which is an open problem), a(n) will become 0.

N. J. A. Sloane, Table of n, a(n) for n = 0..2390

A323797 a(n)..a(n+1)-1 are the indices in A323796 pointing to A006960(n) in the Reverse and Add! graph with 196 seed.

Original entry on oeis.org

1, 9, 13, 55, 103, 143, 183, 191, 431, 745, 1393

Offset: 1

A.H.M. Smeets, Jan 28 2019

This is a supporting sequence for A323796.

Cf. A006960, A323796.

A011375 Length of n-th term in A006960.

Original entry on oeis.org

3, 3, 4, 4, 5, 5, 5, 6, 7, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 11, 11, 12, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 16, 17, 17, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29

Offset: 0

Simon Plouffe

Fred Gruenberger, How to handle numbers with thousands of digits, and why one might want to, Computer Recreations, Scientific American, 250 (No. 4, 1984), 19-26.

A056964 a(n) = n + reversal of digits of n.

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 33, 44, 55, 66, 77, 88, 99, 110, 121, 132, 44, 55, 66, 77, 88, 99, 110, 121, 132, 143, 55, 66, 77, 88, 99, 110, 121, 132, 143, 154, 66, 77, 88, 99, 110

Offset: 0

Henry Bottomley, Jul 18 2000

If n has an even number of digits then a(n) is a multiple of 11.

Also called the Reverse and Add!, or RADD operation. Iteration of this function leads to the definition of Lychrel and related numbers, cf. A023108, A063048, A088753, A006960, and many others. - M. F. Hasler, Apr 13 2019

			a(17) = 17 + 71 = 88.

Indranil Ghosh, Table of n, a(n) for n = 0..20000 (first 1001 terms from T. D. Noe)
Eric Weisstein's World of Mathematics, Reverse-Then-Add Sequence
Index entries for sequences related to Reverse and Add!

Cf. A004086, A056965, A067030.
Differs from A052008 when n=101 and a(101)=202 while A052008(101)=121
Cf. A036839.

a056964 n = n + a004086 n  -- Reinhard Zumkeller, Oct 14 2011

Table[n+FromDigits[Reverse[IntegerDigits[n]]],{n,0,100}] (* Harvey P. Dale, Jul 19 2014 *)
#+IntegerReverse[#]&/@Range[0,100] (* Harvey P. Dale, Jul 31 2025 *)

A056964(n)=fromdigits(Vecrev(digits(n)))+n \\ Charles R Greathouse IV, Oct 28 2014

def A056964(n): return n+int(str(n)[::-1]) # Indranil Ghosh, Jan 29 2017

a(n) = n + A004086(n) = 2*n - A056965(n).

n < a(n) < 11n for n > 0. - Charles R Greathouse IV, Nov 17 2022

A023108 Positive integers which apparently never result in a palindrome under repeated applications of the function A056964(x) = x + (x with digits reversed).

Original entry on oeis.org

196, 295, 394, 493, 592, 689, 691, 788, 790, 879, 887, 978, 986, 1495, 1497, 1585, 1587, 1675, 1677, 1765, 1767, 1855, 1857, 1945, 1947, 1997, 2494, 2496, 2584, 2586, 2674, 2676, 2764, 2766, 2854, 2856, 2944, 2946, 2996, 3493, 3495, 3583, 3585, 3673, 3675

Offset: 1

David W. Wilson

196 is conjectured to be smallest initial term which does not lead to a palindrome. John Walker, Tim Irvin and others have extended this to millions of digits without finding one (see A006960).
Also called Lychrel numbers, though the definition of "Lychrel number" varies: Purists only call the "seeds" or "root numbers" Lychrel; the "related" or "extra" numbers (arising in the former's orbit) have been coined "Kin numbers" by Koji Yamashita. There are only 2 "root" Lychrels below 1000 and 3 more below 10000, cf. A088753. - M. F. Hasler, Dec 04 2007
Question: when do numbers in this sequence start to outnumber numbers that are not in the sequence? - J. Lowell, May 15 2014
Answer: according to Doucette's site, 10-digit numbers have 49.61% of Lychrels. So beyond 10 digits, Lychrels start to outnumber non-Lychrels. - Dmitry Kamenetsky, Oct 12 2015
From the current definition it is unclear whether palindromes are excluded from this sequence, cf. A088753 vs A063048. 9999 would be the first palindromic term that will never result in a palindrome when the Reverse-then-add function A056964 is repeatedly applied. - M. F. Hasler, Apr 13 2019

			From _M. F. Hasler_, Feb 16 2020: (Start)
Under the "add reverse" operation, we have:
196 (+ 691) -> 887 (+ 788) -> 1675 (+ 5761) -> 7436 (+ 6347) -> 13783 (+ 38731) -> etc. which apparently never leads to a palindrome.
Similar for 295 (+ 592) -> 887, 394 (+ 493) -> 887, 790 (+ 097) -> 887 and 689 (+ 986) -> 1675, which all merge immediately into the above sequence, and also for the reverse of any of the numbers occurring in these sequences: 493, 592, 691, 788, ...
879 (+ 978) -> 1857 -> 9438 -> 17787 -> 96558 is the only other "root" Lychrel below 1000 which yields a sequence distinct from that of 196. (End)

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

A.H.M. Smeets, Table of n, a(n) for n = 1..20000 (tested for 200 iterations; first 249 terms from William Boyles)
DeCode, Lychrel Number, dCode.fr 'toolkit' to solve games, riddles, geocaches, 2020.
Jason Doucette, World Records
Martianus Frederic Ezerman, Bertrand Meyer and Patrick Sole, On Polynomial Pairs of Integers, arXiv:1210.7593 [math.NT], 2012-2014.
Patrick De Geest, Some thematic websources
James Grime and Brady Haran, What's special about 196?, Numberphile video (2015).
Fred Gruenberger, How to handle numbers with thousands of digits, and why one might want to, Computer Recreations, Scientific American, 250 (No. 4, 1984), 19-26.
R. K. Guy, What's left?, Math Horizons, Vol. 5, No. 4 (April 1998), pp. 5-7.
Tim Irvin, About Two Months of Computing, or An Addendum to Mr. Walker's Three Years of Computing
Niphawan Phoopha and Prapanpong Pongsriiam, Notes on 1089 and a Variation of the Kaprekar Operator, Int'l J. Math. Comp. Sci. (2021) Vol. 16, No. 4, 1599-1606.
Project Euler, Problem 55: How many Lychrel numbers are there below ten-thousand?
A.H.M. Smeets, Distribution of terms < 10000000 (number of terms in interval of length 10000)
Wade VanLandingham, 196 and other Lychrel numbers
Wade VanLandingham, Largest known Lychrel number
John Walker, Three Years Of Computing: Final Report On The Palindrome Quest
Eric Weisstein's World of Mathematics, 196 Algorithm.
Eric Weisstein's World of Mathematics, Palindromic Number Conjecture
Eric Weisstein's World of Mathematics, Lychrel Number
Index entries for sequences related to Reverse and Add!

Cf. A006960, A088753, A063048, A089694, A089521, A023109; A075421, A030547.

Cf. A056964 ("reverse and add" operation on which this is based).

With[{lim = 10^3}, Select[Range@ 4000, Length@ NestWhileList[# + IntegerReverse@ # &, #, ! PalindromeQ@ # &, 1, lim] == lim + 1 &]] (* Michael De Vlieger, Dec 23 2017 *)

select( {is_A023108(n, L=exponent(n+1)*5)=while(L--&& n*2!=n+=A004086(n),);!L}, [1..3999]) \\ with {A004086(n)=fromdigits(Vecrev(digits(n)))}; default value for search limit L chosen according to known records A065199 and indices A065198. - M. F. Hasler, Apr 13 2019, edited Feb 16 2020

Edited by M. F. Hasler, Dec 04 2007

A033865 Start with n; if palindrome, stop; otherwise add to itself with digits reversed; a(n) gives palindrome at which it stops, or -1 if no palindrome is ever reached.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 11, 33, 44, 55, 66, 77, 88, 99, 121, 22, 33, 22, 55, 66, 77, 88, 99, 121, 121, 33, 44, 55, 33, 77, 88, 99, 121, 121, 363, 44, 55, 66, 77, 44, 99, 121, 121, 363, 484, 55, 66, 77, 88, 99, 55, 121, 363, 484, 1111, 66, 77, 88, 99, 121, 121, 66, 484, 1111, 4884, 77, 88, 99, 121, 121, 363, 484, 77, 4884, 44044, 88

Offset: 0

David W. Wilson

It is believed that a(196) = -1.

			19 -> 19 + 91 = 110 -> 110 + 011 = 121, so a(19) = 121.

M. Donner, I Love Me, Vol. I: S. Wordrow's palindromic encyclopedia (Algonquin Books, 1996) p. 268

T. D. Noe, Table of n, a(n) for n = 0..195
O. Forster, ARIBAS
Mathforum, Making Numbers into Palindromic Numbers
Eric Weisstein's World of Mathematics, 196-Algorithm.

Cf. A061563, A016016, A023109, A006960, A023108, A002113, A033665 (number of steps).

var st: stack; end; for k := 0 to 60 do n := k; while n <> int_reverse(n) do n := n + int_reverse(n); end; stack_push(st,n); end; stack2array(st);

Table[NestWhile[# + FromDigits[Reverse[IntegerDigits[#]]] &, n, IntegerDigits[#] != Reverse[IntegerDigits[#]] &], {n, 0, 90}] (* Harvey P. Dale, Dec 18 2011 *)

a(n)=my(k); while((k=fromdigits(Vecrev(digits(n)))) != n, n += k); n \\ infinite loop if a(n) = -1; Charles R Greathouse IV, Dec 13 2015

More terms from Jenise Smalley (neicey01(AT)hotmail.com), Oct 18 2001

A023109 a(0) = 0. For n > 0, smallest non-palindromic number k such that the smallest palindrome in the Reverse and Add! trajectory of k is reached after exactly n iterations.

Original entry on oeis.org

0, 10, 19, 59, 69, 166, 79, 188, 193, 1397, 829, 167, 2069, 1797, 849, 177, 1496, 739, 1798, 10777, 6999, 1297, 869, 187, 89, 10797, 10853, 10921, 10971, 13297, 10548, 13293, 17793, 20889, 700269, 106977, 108933, 80359, 13697, 10794, 15891, 1009227, 1007619, 1009246, 1008628, 600259, 131996, 70759, 1007377, 1001699, 600279, 141996, 70269, 10677, 10833, 10911

Offset: 0

David W. Wilson

From Felix Fröhlich, May 28 2022: (Start)
Variant of A015994 not allowing palindromes as starting values.
Smallest non-palindromic k such that A033665(k) = n. (End)

Chai Wah Wu, Table of n, a(n) for n = 0..100
Jason Doucette, World Records
Eric Weisstein's World of Mathematics, 196-Algorithm.
Index entries for sequences related to Reverse and Add!

Cf. A015994, A033665, A006960.

Table[ SelectFirst[Range[0, 20000], (np = #; i = 0;
    While[ ! PalindromeQ[np] && i <= n, np = np + IntegerReverse[np];
     i++]; i == n ) &] , {n, 0, 32}] (* Robert Price, Oct 16 2019 *)

rev(n)={d=digits(n);p="";for(i=1,#d,p=concat(Str(d[i]),p));return(eval(p))}
nbs(n)=if(n==rev(n),return(0));for(k=1,10^3,i=n+rev(n);if(rev(i)==i,return(k));n=i) \\ A033665
a(n)=for(k=1,10^8,if(nbs(k)==n,return(k)))
n=0;while(n<100,print1(a(n),", ");n++) \\ Derek Orr, Jul 28 2014

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

def A023109(n):
    if n > 0:
        k = 0
        while True:
            m = k
            for i in range(n):
                if str(m) == str(m)[::-1]:
                    break
                m += int(str(m)[::-1])
            else:
                if str(m) == str(m)[::-1]:
                    return k
            k += 1
    else:
        return 0
# Chai Wah Wu, Feb 08 2015

a(41)-a(55) verified and added by Aldo González Lorenzo, May 15 2011

Name edited by Felix Fröhlich, May 28 2022

cp's OEIS Frontend

A331556 The lower (or left) offset of a 196-iterate (A006960) from the largest palindrome less than the iterate.

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A331557 The upper (or right) offset of a 196-iterate (A006960) from the smallest palindrome greater than the iterate.

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A331560 Size of the palindrome-free intervals about the 196-iterates, A006960.

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A217774 Palindromic deviation of terms of A006960.

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A323797 a(n)..a(n+1)-1 are the indices in A323796 pointing to A006960(n) in the Reverse and Add! graph with 196 seed.

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A011375 Length of n-th term in A006960.

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A056964 a(n) = n + reversal of digits of n.

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A023108 Positive integers which apparently never result in a palindrome under repeated applications of the function A056964(x) = x + (x with digits reversed).

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A033865 Start with n; if palindrome, stop; otherwise add to itself with digits reversed; a(n) gives palindrome at which it stops, or -1 if no palindrome is ever reached.