cp's OEIS Frontend

A088753 Numbers k such that the Reverse and Add! trajectory of k (presumably) does not reach a palindrome (with the exception of k itself) and does not join the trajectory of any term m < k.

%I A088753 #21 Aug 19 2025 18:04:40
%S A088753 196,879,1997,7059,9999,10553,10563,10577,10583,10585,10638,10663,
%T A088753 10668,10697,10715,10728,10735,10746,10748,10783,10785,10787,10788,
%U A088753 10877,10883,10963,10965,10969,10977,10983,10985,12797,12898,13097,13197,13694,14096,14698,15297,15597,18598,18798
%N A088753 Numbers k such that the Reverse and Add! trajectory of k (presumably) does not reach a palindrome (with the exception of k itself) and does not join the trajectory of any term m < k.
%C A088753 Although the starting number k is regarded as part of the trajectory, it is allowed to be palindromic. Hence palindromes are not excluded from the sequence. A063048 is obtained if palindromes are excluded. The smallest term in A088753 but not in A063048 is 9999, the smallest term in A063048 but not in A088753 is 19098.
%C A088753 W. VanLandingham and others have computed nearly 10^7 terms (all terms < 10^14), cf. W. VanLandingham, 196 and Other Lychrel Numbers.
%C A088753 From _M. F. Hasler_, Apr 13 2019: (Start)
%C A088753 Lychrel numbers listed here are also called "seeds", in contrast to Kin numbers A023108 which include all terms in the orbits of the former.
%C A088753 It is not easy to determine whether the orbit of a given term will never merge into the orbit of an earlier term. It seems that the property of "disjoint orbit" is as conjectural as the property of not reaching a palindrome. One could specify a "search limit" in order to get a well-defined sequence. The given list of terms has been checked and extended by considering the orbits up to members of size <= 10^199 at least. Given that the number increases by a factor 10 roughly every 2.416 iterations, this corresponds to about 500 iterations. (End)
%H A088753 M. F. Hasler, <a href="/A088753/b088753.txt">Table of n, a(n) for n = 1..74</a> (all terms up to 10^5), Apr 13 2019.
%H A088753 W. VanLandingham, <a href="http://web.archive.org/www.p196.org/">196 and Other Lychrel Numbers</a> [Copy on web.archive.org as of 05/2018, original site p196.org seems no longer available. - _M. F. Hasler_, Apr 13 2019]
%e A088753 From _M. F. Hasler_, Apr 13 2019: (Start)
%e A088753 All numbers < 196 quickly reach a palindrome under iterations of the reverse-and-add function A056964, cf. A033665.
%e A088753 a(1) = 196 is the smallest integer which appears to never reach a palindrome (checked up to 10^9 iterations!).
%e A088753 Next, A056964(196) = 196 + 691 = 887 is in the orbit of 196 and will therefore never reach a palindrome if 196 does not. However, we do not list this term in this sequence because it is in the orbit of the smaller term 196.
%e A088753 Similarly, 295 + 592 = 887 = A056964(196). Therefore, 295 will also never reach a palindrome if 196 (and therefore 887) doesn't. But again we will not list this number, because its orbit merges into that of the smaller term 196.
%e A088753 The next number which appears to be a Lychrel and has an orbit (conjectured to be) disjoint with that of 196 is 897 = a(2). (End)
%t A088753 limit = 10^3; (* Assumes that there is no palindrome if none is found before "limit" iterations *)
%t A088753 utraj = {};
%t A088753 Select[Range[0,
%t A088753   20000], (np = # + IntegerReverse[#];
%t A088753    x = NestWhileList[ # + IntegerReverse[#] &, np, ! PalindromeQ[#] &, 1, limit];
%t A088753    If[Length[x] >= limit  && Intersection[x, utraj] == {},
%t A088753     utraj = Union[utraj, x, {np}]; True,
%t A088753 utraj = Union[utraj, x, {np}]]) &] (* _Robert Price_, Oct 16 2019 *)
%o A088753 (PARI) A088753_upto(LIM=2e4,M=1e199)={my(U=[],a=List());for(n=1,LIM, my(t=n); while( t<M, is_A002113(t=A056964(t)) && next(2)); setsearch(U,t) && next; U=setunion(U,[t]); print1(n","); listput(a,n)); Set(a)} \\ _M. F. Hasler_, Apr 13 2019
%Y A088753 Cf. A063048 (variant excluding palindromes), A023108 (Kin numbers), A056964 (reverse-and-add), A006960 (orbit of 196), A033665 (steps to reach a palindrome), A061563 (terminating palindrome of n's orbit), A002113 (palindromes).
%K A088753 base,nonn
%O A088753 1,1
%A A088753 _Klaus Brockhaus_, Nov 04 2003
%E A088753 Edited by _M. F. Hasler_, Apr 13 2019