Search: a089694 -id:a089694
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A023108
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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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+10
71
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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
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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Daniel Lignon, Dictionnaire de (presque) tous les nombres entiers, Ellipses, Paris, 2012, 702 pages. See Entry 196.
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LINKS
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DeCode, Lychrel Number, dCode.fr 'toolkit' to solve games, riddles, geocaches, 2020.
R. K. Guy, What's left?, Math Horizons, Vol. 5, No. 4 (April 1998), pp. 5-7.
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EXAMPLE
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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)
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MATHEMATICA
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With[{lim = 10^3}, Select[Range@ 4000, Length@ NestWhileList[# + IntegerReverse@ # &, #, ! PalindromeQ@ # &, 1, lim] == lim + 1 &]] (* Michael De Vlieger, Dec 23 2017 *)
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PROG
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(PARI) 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
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CROSSREFS
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Cf. A056964 ("reverse and add" operation on which this is based).
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KEYWORD
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nonn,base,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A243238
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Table T(n,r) of terms in the reverse and add sequences of positive integers n read by antidiagonals.
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+10
11
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1, 2, 2, 4, 4, 3, 8, 8, 6, 4, 16, 16, 12, 8, 5, 77, 77, 33, 16, 10, 6, 154, 154, 66, 77, 11, 12, 7, 605, 605, 132, 154, 22, 33, 14, 8, 1111, 1111, 363, 605, 44, 66, 55, 16, 9, 2222, 2222, 726, 1111, 88, 132, 110, 77, 18, 10, 4444, 4444, 1353, 2222, 176, 363, 121, 154, 99, 11, 11
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OFFSET
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1,2
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LINKS
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EXAMPLE
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T(5,6) = 88, since 88 is the 6th term in the reverse and add sequence of 5.
Table starts with:
1 2 4 8 16 77 154 605 1111 2222
2 4 8 16 77 154 605 1111 2222 4444
3 6 12 33 66 132 363 726 1353 4884
4 8 16 77 154 605 1111 2222 4444 8888
5 10 11 22 44 88 176 847 1595 7546
6 12 33 66 132 363 726 1353 4884 9768
7 14 55 110 121 242 484 968 1837 9218
8 16 77 154 605 1111 2222 4444 8888 17776
9 18 99 198 1089 10890 20691 40293 79497 158994
10 11 22 44 88 176 847 1595 7546 14003
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MAPLE
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T:= proc(n, r) option remember; `if`(r=1, n, (h-> h +(s->
parse(cat(s[-i]$i=1..length(s))))(""||h))(T(n, r-1)))
end:
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MATHEMATICA
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rad[n_] := n + FromDigits[Reverse[IntegerDigits[n]]];
T[n_, 1] := n; T[n_, k_] := T[n, k] = rad[T[n, k-1]];
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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9999, 99999, 990099, 999999, 9901099, 9905099, 9993999, 9996999, 9997999, 9998999, 9999999, 99999999, 990959099, 990969099, 999010999, 999020999, 999030999, 999040999, 999070999, 999929999, 999939999, 999969999, 999989999
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OFFSET
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1,1
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COMMENTS
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Palindromes in A088753; palindromes for which the Reverse and Add! process does not lead to another palindrome. The numbers were extracted from W. VanLandingham's list of Lychrel numbers.
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LINKS
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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