A075421 -id:A075421 - cp's OEIS frontend

A091676 a(n) = smallest k such that the base 4 Reverse and Add! trajectory of A075421(n) joins the trajectory of k.

266, 3, 719, 795, 799, 269, 258, 286, 4207, 1037, 4236, 4278, 256, 4169, 4182, 4189, 271, 4338, 4402, 4598, 4662, 4108, 312, 5357, 6157, 4104, 4159, 7247, 7295, 7407, 7549, 8063, 4157, 8189, 4141, 12431, 12463, 12539, 15487, 4349, 4239, 7391, 16522

Offset: 1

Klaus Brockhaus, Jan 28 2004

a(n) <= A075421(n); a(n) = A075421(n) iff the trajectory of A075421(n) does not join the trajectory of any smaller number, i.e., A075421(n) is also a term of A091675.
a(n) determines a 1-1-mapping from the terms of A075421 to the terms of A091675. For the inverse mapping cf. A091677.
Base-4 analog of A089493.

			A075421(1) = 290, the trajectory of 290 (A075299) joins the trajectory of 266 = A091675(12) at 4195, so a(1) = 266. A075421(6) = 1210, the trajectory of 1210 joins the trajectory of 269 = A091675(13) at 17975, so a(6) = 269.

Index entries for sequences related to Reverse and Add!

Cf. A075421, A091675, A075299, A089493.

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

A063048 Numbers n such that the Reverse and Add! trajectory of n (presumably) does not reach a palindrome and does not join the trajectory of any term m < n.

Original entry on oeis.org

196, 879, 1997, 7059, 10553, 10563, 10577, 10583, 10585, 10638, 10663, 10668, 10697, 10715, 10728, 10735, 10746, 10748, 10783, 10785, 10787, 10788, 10877, 10883, 10963, 10965, 10969, 10977, 10983, 10985, 12797, 12898, 13097, 13197, 13694

Offset: 1

Klaus Brockhaus, Jul 07 2001, revised Nov 04 2003

The starting number n is regarded as part of the trajectory, so palindromes are excluded from the sequence. A088753 is obtained if palindromes are not excluded. The smallest term in A063048 but not in A088753 is 19098, the smallest term in A088753 but not in A063048 is 9999.
Subsequence of A023108. Sequence A070788 is similarly defined, but palindromes are irrelevant. Corresponding sequences for other bases are A075252 (base 2), A077405 (base 3), A075421 (base 4).
If the trajectory of a number k joins the trajectory of a smaller number which is a term of the present sequence, then this occurs after very few Reverse and Add! steps (at most 8 for k < 100000, at most 10 for k < 1000000). On the other hand, the trajectories of the terms < 14000 do not join the trajectory of any smaller term within at least 1500 steps. This is the precise meaning of "presumably" in the definition.
The terms are rather unevenly distributed. They form clusters, especially above 10^4, 10^5, 10^6, ... . The interval from 10000 to 11000 for example contains 26 terms, whereas only two terms occur in the interval from 90000 to 100000.
It seems that if the most significant digit is not equal to 1, the least significant digit is always 9, while this does not hold for the Lychrel numbers as in A023108. - A.H.M. Smeets, Feb 18 2019
From A.H.M. Smeets, Sep 18 2021: (Start)
Let d_0 d_1 d_2 ... d_n be the decimal digits of an (n+1)-digit number.
All numbers in this sequence seem to satisfy the following condition:
d_0 = "1" or d_n = "9", and for all k, 0 < k < floor((n-1)/2), d_k = "0" or d_k = "9" or d_(n-k) = "0" or d_(n-k) = "9".
The plot log_10(a(n)) versus log_10(n) shows a stepwise behavior. However, the global behavior seems to be a straight line with slope e/(e-1) (= A185393). This slope is also obtained for the seeds in the Reverse and Add! problem in other bases. (End)

			1997 is a term since the trajectory of 1997 (presumably) does not lead to a number which occurs in the trajectory of 196 or of 879 (actually checked for the first 10000 terms of these trajectories). The trajectory of 1997 joins the trajectory of 106 at 97768 (cf. A070796), but 106 is not a term of the present sequence.

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..21491 (terms 1..1711 from Klaus Brockhaus).
Klaus Brockhaus, Illustration: Distribution of terms below 2000000
Klaus Brockhaus, List of terms below 2000000
A.H.M. Smeets, log_10(a(n)) versus log_10(n)
Index entries for sequences related to Reverse and Add!

Cf. A033865, A023108, A006960, A063049 - A063065, A088753, A070788, A075252, A077405, A075421, A070796.

limit = 10^3; (* Assumes that there is no palindrome if none is found before "limit" iterations *)
utraj = {};
Select[Range[0, 14000], (x = NestWhileList[ # + IntegerReverse[#] &, #, ! PalindromeQ[#] &, 1, limit];
   If[Length[x] >= limit && Intersection[x, utraj] == {},
    utraj = Union[utraj, x]; True,
utraj = Union[utraj, x]]) &] (* Robert Price, Oct 16 2019 *)

A075252 Trajectory of n under the Reverse and Add! operation carried out in base 2 does not reach a palindrome and (presumably) does not join the trajectory of any term m < n.

Original entry on oeis.org

22, 77, 442, 537, 775, 1066, 1081, 1082, 1085, 1115, 1562, 1575, 1587, 2173, 3355, 3599, 3871, 4099, 4153, 4185, 4193, 4202, 4262, 4285, 4402, 4633, 4666, 6163, 6166, 6374, 9241, 9466, 16544, 16546, 16586, 16601, 16613, 16616, 16720, 16748, 16994

Offset: 1

Klaus Brockhaus, Sep 10 2002

Base-2 analog of A063048 (base 10) and A075421 (base 4); subsequence of A066059. - For the trajectory of 22 (cf. A061561) and the trajectory of 77 (cf. A075253) it has been proved that they do not contain a palindrome. A similar proof can be given for most terms of this sequence, but there are a few terms (4262, 17498, 33378, 33898, ...) whose trajectory does not show the kind of regularity that can be utilized for the construction of a proof. - If the trajectory of an integer k joins the trajectory of a smaller integer which is a term of the present sequence, then this occurs after very few 'Reverse and Add!' steps (at most 84 for numbers < 20000). On the other hand, the trajectories of the terms of this sequence do not join the trajectory of any smaller term within at least 1000 steps.
From A.H.M. Smeets, Feb 12 2019: (Start)
Most terms in this sequence eventually give rise to a regular binary pattern. These regular patterns can be represented by contextfree grammars:
S_a -> 10 T_a 00, T_a -> 1 T_a 0 | A_a(n);
S_b -> 11 T_b 01, T_b -> 0 T_b 1 | B_a(n);
S_c -> 10 T_c 000, T_c -> 1 T_c 0 | C_a(n) and
S_d -> 11 T_d 101, T_d -> 0 T_d 1 | D_a(n).
A_22 = 1101, B_22 = 1000, C_22 = 1101, D_22 = 0010 (see also A058042);
A_77 = 1100010, B_77 = 0000101, C_77 = 1101011, D_77 = 0100000 (see also A075253)
Decimal representations for 10 A_a(n) 00 are given by A306514(n).
Binary representations for 10 A_a(n) 00 are given by A306515(n).
(End)

			442 is a term since the trajectory of 442 (presumably) does not lead to an integer which occurs in the trajectory of 22 or of 77.

Klaus Brockhaus, On the 'Reverse and Add!' algorithm in base 2
Index entries for sequences related to Reverse and Add!

Cf. A063048, A075421, A066059, A058042, A061561, A075253, A306514, A306515.

limit = 10^2; (* Assumes that there is no palindrome if none is found before "limit" iterations *)
utraj = {};
Select[Range[0, 17000], (x = NestWhileList[# + IntegerReverse[#, 2] &, #, # != IntegerReverse[#, 2] & , 1, limit];
   If[Length[x] >= limit  && Intersection[x, utraj] == {},
    utraj = Union[utraj, x]; True,
utraj = Union[utraj, x]]) &] (* Robert Price, Oct 16 2019 *)

A075153 Trajectory of 318 under the Reverse and Add! operation carried out in base 4, written in base 10.

Original entry on oeis.org

318, 1071, 5040, 5985, 10710, 20400, 24225, 43350, 81600, 85425, 165750, 327360, 342705, 664950, 1309440, 1324785, 2629110, 5241600, 5303025, 10524150, 20966400, 21027825, 41973750, 83880960, 84126705, 167925750, 335523840

Offset: 0

Klaus Brockhaus, Sep 05 2002

290 is conjectured (cf. A066450) to be the smallest number such that the Reverse and Add! algorithm in base 4 does not lead to a palindrome. 318 (not 255 since 255 is a base 4 palindrome) is up to now the smallest number whose base 4 trajectory provably does not contain a palindrome. A proof along the lines of Klaus Brockhaus, On the 'Reverse and Add!' algorithm in base 2, can be based on the formula given below.
lim_{n -> infinity} a(n)/a(n-1) = 2 for n mod 3 in {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = 1 for n mod 3 = 0.

			318 (decimal) = 10332 -> 10332 + 23301 = 100233 = 1071 (decimal).

Klaus Brockhaus, On the 'Reverse and Add!' algorithm in base 2
K. S. Brown, Digit Reversal Sums Leading to Palindromes
Index entries for sequences related to Reverse and Add!

Cf. A058042 (trajectory of binary number 10110 (decimal 22)), A061561 (A058042 written in base 10), A066450 (conjectured minimal k so that the trajectory of k in base n does not lead to a palindrome).
Cf. A075253 (trajectory of 77 in base 2), A075420 (trajectory of n in base 4 (presumably) does not reach a palindrome), A075421 (trajectory of n in base 4 (presumably) does not reach a palindrome and (presumably) does not join the trajectory of any term m < n), A075299 (trajectory of 290 in base 4), A075466 (trajectory of 266718 in base 4), A075467 (trajectory of 270798 in base 4), A076247 (trajectory of 1059774 in base 4), A076248 (trajectory of 1059831 in base 4), A091675 (trajectory of n in base 4 (presumably) does not join the trajectory of any m < n).
Cf. A166912 (a(6*n)/3), A166913 (a(6*n+1)/3), A166914 (a(6*n+2)/240), A166915 (a(6*n+3)/15), A166916 (a(6*n+4)/30), A166917 (a(6*n+5)/240).

trajectory:=function(init, steps, base) a:=init; S:=[a]; for n in [1..steps] do a+:=Seqint(Reverse(Intseq(a,base)),base); Append(~S, a); end for; return S; end function; trajectory(318, 26, 4);

NestWhileList[# + IntegerReverse[#, 4] &, 318,  # !=
IntegerReverse[#, 4] &, 1, 26] (* Robert Price, Oct 18 2019 *)

{m=318; stop=29; c=0; while(c0,d=divrem(k,4); k=d[1]; rev=4*rev+d[2]); c++; m=m+rev)}

a(0) = 318; a(1) = 1071; for n > 1 and n = 2 (mod 6): a(n) = 5*4^(2*k+5)-5*4^(k+2) where k = (n-2)/6; n = 3 (mod 6): a(n) = 5*4^(2*k+5)+55*4^(k+2)-15 where k = (n-3)/6; n = 4 (mod 6): a(n) = 10*4^(2*k+5)+30*4^(k+2)-10 where k = (n-4)/6; n = 5 (mod 6): a(n) = 20*4^(2*k+5)-5*4^(k+2) where k = (n-5)/6; n = 0 (mod 6): a(n) = 20*4^(2*k+5)+235*4^(k+2)-15 where k = (n-6)/6; n = 1 (mod 6): a(n) = 40*4^(2*k+5)+150*4^(k+2)-10 where k = (n-7)/6.

G.f.: 3*(106 +357*x +1680*x^2 +1465*x^3 +1785*x^4 -1600*x^5 -1900*x^6 -3400*x^7 -6800*x^8 -9780*x^9 -9860*x^10 +6720*x^11 +10064*x^12 +11088*x^13) / ((1-x)*(1+x+x^2)*(1-2*x^3)*(1+2*x^3)*(1-4*x^3)).

Two comments added, g.f. edited, MAGMA program and cross-references added by Klaus Brockhaus, Oct 26 2009

A075420 Trajectory of n under the Reverse and Add! operation carried out in base 4 (presumably) does not reach a palindrome.

Original entry on oeis.org

290, 318, 378, 381, 438, 444, 462, 498, 501, 504, 510, 545, 567, 573, 627, 633, 636, 639, 693, 696, 699, 717, 719, 732, 751, 753, 756, 759, 765, 775, 795, 799, 800, 822, 823, 828, 835, 847, 859, 882, 883, 888, 894, 895, 915, 919, 927, 948, 954, 967, 972

Offset: 1

Klaus Brockhaus, Sep 18 2002

Base-4 analog of A023108 (base 10) and A066059 (base 2).

A.H.M. Smeets, Table of n, a(n) for n = 1..20000
Index entries for sequences related to Reverse and Add!

Cf. A023108, A066059, A075421, A075299, A075153.

limit = 10^3; (* Assumes that there is no palindrome if none is found before "limit" iterations *)
Select[Range[1000],
Length@NestWhileList[# + IntegerReverse[#, 4] &, #, # !=
IntegerReverse[#, 4]  &, 1, limit] == limit + 1 &] (* Robert Price, Oct 16 2019 *)

{stop=1000; for(n=1,980,k=n; c=0; while(c0,d=divrem(a,4); a=d[1]; rev=4*rev+d[2]); if(rev==k,c=stop+1,k=k+rev; c++)); if(c==stop,print1(n,",")))}

Offset changed to 1 by A.H.M. Smeets, Feb 10 2019

A077405 Trajectory of n under the Reverse and Add! operation carried out in base 3 (presumably) does not reach a palindrome and (presumably) does not join the trajectory of any term m < n.

Original entry on oeis.org

103, 746, 805, 2231, 2326, 2671, 2725, 2959, 2969, 3679, 4421, 4430, 4439, 4448, 5894, 6626, 6638, 6686, 6698, 6733, 6741, 6779, 6789, 6793, 6943, 7124, 7365, 7849, 8093, 8801, 8836, 10771, 11078, 11158, 13184, 13361, 17558, 17639, 19115, 19196, 19733, 19895, 19901, 19907, 20106, 20124, 20149, 20161

Offset: 1

Klaus Brockhaus, Nov 05 2002

Base 3 analog of A075252 (base 2), A075421 (base 4) and A063048 (base 10); subsequence of A077404. - A proof that the base 3 trajectory does not contain a palindrome has been found up to now for none of the terms. - If the trajectory of an integer k joins the trajectory of a smaller integer which is a term of the present sequence, then this occurs after very few Reverse and Add! steps (at most 9 for k < 20000). On the other hand, the trajectories of the terms of this sequence do not join the trajectory of any smaller term within at least 1000 steps.

			805 is a term since the trajectory of 805 (presumably) does not lead to a number which occurs in the trajectory of 103 or of 746.

A.H.M. Smeets, Table of n, a(n) for n = 1..529
Index entries for sequences related to Reverse and Add!

Cf. A063048, A075252, A075421, A077404.

limit = 10^3; (* Assumes that there is no palindrome if none is found before "limit" iterations *)
utraj = {};
Select[Range[0,21000], (i = 0;
   x = NestWhileList[(i++; # + IntegerReverse[#, 3]) &, #, # !=
        IntegerReverse[#, 3] && i < limit  &];
   If[i >= limit  && Intersection[x, utraj] == {},
    utraj = Union[utraj, x]; True,
    utraj = Union[utraj, x]]) &]
(* Robert Price, Oct 19 2019 *)

Offset changed to 1 by A.H.M. Smeets, Feb 14 2019

a(41)-a(48) from A.H.M. Smeets, Feb 18 2019

A075466 Trajectory of 266718 under the Reverse and Add! operation carried out in base 4, written in base 10.

Original entry on oeis.org

266718, 1017375, 2019150, 4934715, 20413980, 34239885, 64220175, 127195950, 321080475, 1286586060, 2154739965, 4288508415, 8571775230, 21401016315, 85781907180, 149736661725, 278082371775, 1369020907200, 1433193762225

Offset: 0

Klaus Brockhaus, Sep 18 2002

266718 = A075421(358) is the smallest term > 318 of A075421 whose base 4 trajectory provably does not contain a palindrome. A proof along the lines of Klaus Brockhaus, On the 'Reverse and Add!' algorithm in base 2, can be based on the formula given below. - The generating function given describes the sequence from a(26) onward; the g.f. for the complete sequence is known but more than twice as big.

			266718 (decimal) = 1001013132 -> 1001013132 + 2313101001 = 3320120133 = 1017375 (decimal).

Klaus Brockhaus, On the 'Reverse and Add!' algorithm in base 2
Index entries for sequences related to Reverse and Add!

Cf. A075153, A075421, A075467.

NestWhileList[# + IntegerReverse[#, 4] &, 266718,  # !=
IntegerReverse[#, 4] &, 1, 23] (* Robert Price, Oct 18 2019 *)

{m=266718; stop=19; c=0; while(c0,d=divrem(k,4); k=d[1]; rev=4*rev+d[2]); c++; m=m+rev)}

a(0), ..., a(18) as above; a(19) = 2780823717750; a(20) = 5492189757120; a(21) = 5749636151985; a(22) = 11156010444150; a(23) = 21968759028480; a(24) = 22226205423345; a(25) = 44109148986870; for n > 25 and n = 2 (mod 6): a(n) = 5*4^(2*k+14)-83865605*4^k where k = (n-2)/6; n = 3 (mod 6): a(n) = 5*4^(2*k+14)+3941683435*4^k-15 where k = (n-3)/6; n = 4 (mod 6): a(n) = 10*4^(2*k+14)+2515968150*4^k-10 where k = (n-4)/6; n = 5 (mod 6): a(n) = 20*4^(2*k+14)-335462420*4^k where k = (n-5)/6; n = 0 (mod 6): a(n) = 20*4^(2*k+14)+3690086620*4^k-15 where k = (n-6)/6; n = 1 (mod 6): a(n) = 40*4^(2*k+14)+2012774520*4^k-10 where k = (n-7)/6. G.f.: -15*(47049901525664*x^11+23708157972464*x^10+23433347158016*x^9-46912496118440*x^8-23502049861628*x^7-23433347158016*x^6-11908468626600*x^5-6137441522940*x^4-5862630708480*x^3+11771063219370*x^2+5931333412095*x+5862630708480)/((x-1)*(x^2+x+1)*(2*x^3-1)*(2*x^3+1)*(4*x^3-1))

A075467 Trajectory of 270798 under the Reverse and Add! operation carried out in base 4, written in base 10.

Original entry on oeis.org

270798, 1005135, 1994670, 5058075, 20047500, 33313725, 66545850, 112201785, 225464610, 368353785, 835135950, 1587633825, 2841028950, 5347819200, 5598498225, 10862757750, 21453946560, 22456662705, 43576370550

Offset: 0

Klaus Brockhaus, Sep 18 2002

The base 4 trajectory of 270798 = A075421(370) provably does not contain a palindrome. A proof along the lines of Klaus Brockhaus, On the 'Reverse and Add!' algorithm in base 2, can be based on the formula given below. - The generating function given describes the sequence from a(11) onward; the g.f. for the complete sequence is known but nearly twice as big.

			270798 (decimal) = 1002013032 -> 1002013032 + 2303102001 = 3311121033 = 1005135 (decimal).

Klaus Brockhaus, On the 'Reverse and Add!' algorithm in base 2
Index entries for sequences related to Reverse and Add!

Cf. A075153, A075421, A075466.

NestWhileList[# + IntegerReverse[#, 4] &, 270798,  # !=
IntegerReverse[#, 4] &, 1, 23] (* Robert Price, Oct 18 2019 *)

{m=270798; stop=20; c=0; while(c0,d=divrem(k,4); k=d[1]; rev=4*rev+d[2]); c++; m=m+rev)}

a(0), ..., a(10) as above; for n > 10 and n = 5 (mod 6): a(n) = 5*4^(2*k+10)+15341035*4^k-15 where k = (n+1)/6; n = 0 (mod 6): a(n) = 10*4^(2*k+10)+9792150*4^k-10 where k = n/6; n = 1 (mod 6): a(n) = 20*4^(2*k+10)-1305620*4^k where k = (n-1)/6; n = 2 (mod 6): a(n) = 20*4^(2*k+10)+14361820*4^k-15 where k = (n-2)/6; n = 3 (mod 6): a(n) = 40*4^(2*k+10)+7833720*4^k-10 where k = (n-3)/6; n = 4 (mod 6): a(n) = 80*4^(2*k+10)-1305620*4^k where k = (n-4)/6. G.f.: -15*(1426085120*x^11+749251744*x^10+419191024*x^9-1430263104*x^8-715827880*x^7-369055228*x^6-352343296*x^5-222825800*x^4-155978060*x^3+356521280*x^2+189401930*x+105842255)/((x-1)*(x^2+x+1)*(2*x^3-1)*(2*x^3+1)*(4*x^3-1))

A091675 Positive integers n such that the trajectory of n under the Reverse and Add! operation carried out in base 4 (presumably) does not join the trajectory of any m < n.

Original entry on oeis.org

1, 3, 21, 22, 26, 29, 31, 55, 256, 258, 262, 266, 269, 271, 282, 286, 287, 302, 312, 413, 479, 511, 519, 551, 555, 719, 795, 799, 1026, 1029, 1034, 1037, 1066, 1549, 1790, 2863, 3087, 3119, 4096, 4098, 4102, 4104, 4106, 4108, 4109, 4113, 4114, 4116, 4117

Offset: 1

Klaus Brockhaus, Jan 28 2004

The conjecture that the base-4 trajectories of the terms do not join is based on the observation that if the trajectories of two integers below 4120 join, this happens after at most 28 steps, while for any two terms listed above the trajectories do not join within 1000 steps. For pairs from 1, 3, 21, 22, 26, 29, 31, 55 this has even been checked for 5000 steps.

Base-4 analog of A070788.

			The trajectory of 2 is part of the trajectory of 1 (cf. A035524); the trajectory of 3 does not join the trajectory of 1 within 10000 steps; the trajectory of 21 does not join the trajectory of 1 or of 3 within 10000 steps.

Klaus Brockhaus, Illustration: Distribution of terms below 2000000
Index entries for sequences related to Reverse and Add!

Cf. A035524, A075421, A070788.

limit = 10^3; utraj = {};
Select[Range[4120], (x = NestList[ # + IntegerReverse[#, 4] &, #, limit]; If[Intersection[x, utraj] == {}, utraj = Union[utraj, x]; True, utraj = Union[utraj, x]]) &] (* Robert Price, Oct 20 2019 *)

cp's OEIS Frontend

A091676 a(n) = smallest k such that the base 4 Reverse and Add! trajectory of A075421(n) joins the trajectory of k.

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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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A063048 Numbers n such that the Reverse and Add! trajectory of n (presumably) does not reach a palindrome and does not join the trajectory of any term m < n.

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A075252 Trajectory of n under the Reverse and Add! operation carried out in base 2 does not reach a palindrome and (presumably) does not join the trajectory of any term m < n.

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A075153 Trajectory of 318 under the Reverse and Add! operation carried out in base 4, written in base 10.

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A075420 Trajectory of n under the Reverse and Add! operation carried out in base 4 (presumably) does not reach a palindrome.

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A077405 Trajectory of n under the Reverse and Add! operation carried out in base 3 (presumably) does not reach a palindrome and (presumably) does not join the trajectory of any term m < n.

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A075466 Trajectory of 266718 under the Reverse and Add! operation carried out in base 4, written in base 10.