A058042 -id:A058042 - cp's OEIS frontend

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

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))

A062128 In base 2: 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, 11, 11, 101, 101, 1001, 111, 1001, 1001, 1111, 11011, 1111, 11011, 10101, 1111, 10001, 10001, 11011, 1100011, 1100011, 10101, -1, 111111, 11011, 1100011, -1, 11011, -1, 111111, 101101, 11111, 100001, 100001, 110011, -1, 101101, -1, 111111, 1100011, 101101, -1, 111111, 1100011, 1100011

Offset: 0

Klaus Brockhaus, Jun 06 2001

The analog of A033865 in base 2.

			23: 10111 -> 10111 + 11101 = 110100 -> 110100 + 1011 = 111111, so a(23) = 111111.

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

Cf. A033865, A062129, A062130, A058042.

stop := 500; for k := 0 to 60 do c := 0; m := k; rev := bit_reverse(m); while m <> rev and c < stop do inc(c); m := m + rev; rev := bit_reverse(m); end; if c < stop then bit_write(m); else write(-1); end; write(" "); end;

limit = 10^4; (* Assumes that there is no palindrome if none is found before "limit" iterations *)
BaseForm[Table[np = n; i = 0;
  While[np != IntegerReverse[np, 2] && i < limit,
   np = np + IntegerReverse[np, 2]; i++];
If[i >= limit, -1, np], {n, 0, 44}], 2] (* Robert Price, Oct 14 2019 *)

A075299 Trajectory of 290 under the Reverse and Add! operation carried out in base 4, written in base 10.

Original entry on oeis.org

290, 835, 1610, 4195, 17060, 23845, 46490, 89080, 138125, 255775, 506510, 1238395, 5127260, 8616205, 15984335, 31949470, 79793675, 315404860, 569392925, 1060061935, 2114961710, 5206421995, 20997654620, 35262166285

Offset: 0

Klaus Brockhaus, Sep 12 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. Unlike 318 (cf. A075153) its trajectory does not exhibit any recognizable regularity, so that the method by which the base 4 trajectory of 318 as well as the base 2 trajectories of 22 (cf. A061561), 77 (cf. A075253), 442 (cf. A075268) etc. can be proved to be palindrome-free (cf. Links), is not applicable here.

			290 (decimal) = 10202 -> 10202 + 20201 = 31003 = 835 (decimal).

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

Cf. A066450, A075153, A058042, A061561, A075253, A075268.

NestWhileList[# + IntegerReverse[#, 4] &, 290,  # !=
IntegerReverse[#, 4] &, 1, 23] (* Robert Price, Oct 18 2019 *)

{m=290; stop=26; c=0; while(c0,d=divrem(k,4); k=d[1]; rev=4*rev+d[2]); c++; m=m+rev)}

A062129 In base 2: start with n; add to itself with digits reversed; if palindrome, stop; otherwise repeat; a(n) gives palindrome at which it stops, or -1 if no palindrome is ever reached.

Original entry on oeis.org

0, 11, 11, 1001, 101, 1111, 1001, 10101, 1001, 11011, 1111, 11011, 1111, 11011, 10101, 101101, 10001, 110011, 11011, 1100011, 1100011, 111111, -1, 111111, 11011, 1100011, -1, 11111111, -1, 111111, 101101, 1011101, 100001, 1100011, 110011, -1, 101101, -1, 111111, 1100011, 101101, -1, 111111

Offset: 0

Klaus Brockhaus, Jun 06 2001

The analog of A061563 in base 2. Differs from A062128 only for those n, which are palindromes in base 2.

			23: 10111 -> 10111 + 11101 = 110100 -> 110100 + 1011 = 111111, so a(23) = 111111.

Cf. A061563, A062128, A062131, A058042.

stop := 500; for k := 0 to 60 do c := 0; m := k; test := true; while test and c < stop do inc(c); m := m + bit_reverse(m); test := m <> bit_reverse(m); end; if c < stop then bit_write(m); else write(-1); end; write(" "); end;

A066450 a(n) is the conjectured value of the minimal number to which repeated application of the "reverse and add!" algorithm in base n does not terminate in a palindrome. If there is no such number in base n, then a(n) := -1.

Original entry on oeis.org

22, 103, 290, 708, 1079, 2656, 1021, 593, 196, 1011, 237, 2701, 361, 447, 413, 3297, 519, 341, 379, 711, 461, 505, 551, 1022, 649, 701, 755, 811, 869, 929, 991, 1055, 1799, 1922, 1259, 1331, 1405, 1481, 1559, 1639, 1595, 1762, 1891, 1934, 2069, 2161

Offset: 2

Frederick Magata (frederick.magata(AT)uni-muenster.de), Dec 29 2001

It would be nice to remove the word "Conjectured" from the description. - N. J. A. Sloane
All the terms in this sequence except the first are only conjectures. (See Walker, Irvin on a(10)=196 and Brockhaus on a(2)=22.)
An obvious algorithm is: start with r := n and check whether the "reverse and add!" algorithm in base n halts in a palindrome or not. If it stops, increment r by one and repeat the process, else return r. To obtain the values above, an upper limit of 100 "reverse and add!" steps was used.
Conjectures: a(n) shows the same asymptotic behavior as n^2. For infinitely many n, a(n) = n^2 - n - 1. Again, it is an open question, if the values of the sequence really lead to infinitely many "reverse and add!" steps or not. Is the sequence always positive?

Klaus Brockhaus, On the 'Reverse and Add!' algorithm in base 2
T. Irvin, About Two Months of Computing, or An Addendum to Mr. Walker's Three Years of Computing.
J. Walker, Three Years Of Computing: Final Report On The Palindrome Quest
Index entries for sequences related to Reverse and Add!

limit = 10^3; (* Assumes that there is no palindrome if none is found before "limit" iterations *)
Table[SelectFirst[Range[10000],
  Length@NestWhileList[ # + IntegerReverse[#, n] &,  #, # !=
        IntegerReverse[#, n]  &, 1, limit] == limit + 1 &] , {n, 2,
47}] (* Robert Price, Oct 18 2019 *)

David W. Wilson remarks (Jan 02 2002): I verified these using 1000 digits as a stopping point (this would be >>1000 iterations). I am highly confident of these values.

A076247 Trajectory of 1059774 under the Reverse and Add! operation carried out in base 4, written in base 10.

Original entry on oeis.org

1059774, 4187583, 8355006, 20822715, 83391660, 144328605, 268919295, 1339676160, 1349598705, 2683144950, 5361370860, 9358549725, 17380163775, 85563883200, 89574690225, 173801637750, 343262166720, 359352580785

Offset: 0

Klaus Brockhaus, Oct 03 2002

1059774 = A075421(1096) is the fourth term 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(16) onward; the g.f. for the complete sequence is known but nearly twice as big.

			1059774 (decimal) = 10002232332 -> 10002232332 + 23323220001 = 33332112333 = 4187583 (decimal).

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

Cf. A075421, A075153, A075466, A075467, A076248.

NestWhileList[# + IntegerReverse[#, 4] &, 1059774,  # !=
IntegerReverse[#, 4] &, 1, 23] (* Robert Price, Oct 18 2019 *)

{m=1059774; 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(15) as above; for n > 15 and n = 4 (mod 6): a(n) = 5*4^(2*k+12)-5237765*4^k where k = (n+2)/6; n = 5 (mod 6): a(n) = 5*4^(2*k+12)+246174955*4^k-15 where k = (n+1)/6; n = 0 (mod 6): a(n) = 10*4^(2*k+12)+157132950*4^k-10 where k = n/6; n = 1 (mod 6): a(n) = 20*4^(2*k+12)-20951060*4^k where k = (n-1)/6; n = 2 (mod 6): a(n) = 20*4^(2*k+12)+230461660*4^k-15 where k = (n-2)/6; n = 3 (mod 6): a(n) = 40*4^(2*k+12)+125706360*4^k-10 where k = (n-3)/6. G.f.: -15*(185397326496*x^11+95559181296*x^10+91268404224*x^9-183251937960*x^8-92341098492*x^7 -91268404224*x^6-48628806952*x^5-27174921532*x^4-22884144448*x^3+46483418410*x^2 +23956838719*x+22884144448)/((x-1)*(x^2+x+1)*(2*x^3-1)*(2*x^3+1)*(4*x^3-1))

A076248 Trajectory of 1059831 under the Reverse and Add! operation carried out in base 4, written in base 10.

Original entry on oeis.org

1059831, 4728312, 7831065, 14433270, 24913965, 56412450, 92165625, 208908750, 396926625, 710289750, 1336954560, 1398889905, 2715199350, 5363547840, 5614238385, 10894222710, 21453945600, 21701687025, 43073052150

Offset: 0

Klaus Brockhaus, Oct 03 2002

1059831 = A075421(1105 ) is the fifth term 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.

			1059831 (decimal) = 10002233313 -> 10002233313 + 31333220001 = 102002113320 = 4728312 (decimal).

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

Cf. A075421, A075153, A075466, A075467, A076247.

NestWhileList[# + IntegerReverse[#, 4] &, 1059831,  # != IntegerReverse[ #, 4] &, 1, 23] (* Robert Price, Oct 19 2019 *)

{m=1059831; 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(7) as above; for n > 7 and n = 2 (mod 6): a(n) = 5*4^(2*k+9)+3836395*4^k-15 where k = (n+4)/6; n = 3 (mod 6): a(n) = 10*4^(2*k+9)+2450070*4^k-10 where k = (n+3)/6; n = 4 (mod 6): a(n) = 20*4^(2*k+9)-326420*4^k where k = (n+2)/6; n = 5 (mod 6): a(n) = 20*4^(2*k+9)+3544540*4^k-15 where k = (n+1)/6; n = 0 (mod 6): a(n) = 40*4^(2*k+9)+1927800*4^k-10 where k = n/6; n = 1 (mod 6): a(n) = 80*4^(2*k+9)-322580*4^k where k = (n-1)/6. G.f.: -3*(668508000*x^19+444361200*x^18+222142800*x^17-528080680*x^16-356464620*x^15 -125753060*x^14-299532884*x^13-188180432*x^12-143040640*x^11+128992350*x^10+90219415*x^9 +38288125*x^8+28112975*x^7+6666425*x^6+5752375*x^5+424135*x^4+3044705*x^3+2610355*x^2 + 1576104*x+353277)/((x-1)*(x^2+x+1)*(2*x^3-1)*(2*x^3+1)*(4*x^3-1))

A077408 Trajectory of 103 under the Reverse and Add! operation carried out in base 3, written in base 10.

Original entry on oeis.org

103, 230, 436, 776, 2424, 3856, 7400, 20856, 30928, 60920, 220248, 242704, 432896, 857152, 1460408, 2754688, 5134016, 16206744, 24437488, 44623424, 138104472, 201737128, 401511824, 1438324704, 1601682040, 2820726320, 5622321088

Offset: 0

Klaus Brockhaus, Nov 05 2002

103 = A077405(0) is conjectured (cf. A066450) to be the smallest number such that the Reverse and Add! algorithm in base 3 does not lead to a palindrome. Its trajectory does not exhibit any recognizable regularity, so that the method by which the base-2 trajectories of 22 (cf. A061561), 77 (cf. A075253), 442 (cf. A075268) etc. as well as the base-4 trajectories of 318 (cf. A075153), 266718 (cf. A075466), 270798 (cf. A075467) etc. can be proved to be palindrome-free (cf. Links), is not applicable here.

			103 (decimal) = 10211 -> 10211 + 11201 = 22112 = 230 (decimal).

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

Cf. A058042, A077405, A066450, A061561, A075253, A075268, A075153, A075466, A075467.

m := 103; stop := 28; c := 0; while c < stop do write(m:group(0),","); k := m; rev := 0; while k > 0 do rev := 3*rev + (k mod 3); k := k div 3; end; inc(c); m := m+rev; end;

A213012 Trajectory of 26 under the Reverse and Add! operation carried out in base 2.

Original entry on oeis.org

26, 37, 78, 135, 360, 405, 744, 837, 1488, 1581, 3024, 3213, 6048, 6237, 12192, 12573, 24384, 24765, 48960, 49725, 97920, 98685, 196224, 197757, 392448, 393981, 785664, 788733, 1571328, 1574397, 3144192, 3150333

Offset: 0

Ben Branman, Jun 01 2012

26 is the second-smallest number (after 22) whose base 2 trajectory does not contain a palindrome.
lim_{n -> infinity} a(n)/a(n-1) = 2 for n mod 2 = 0.
lim_{n -> infinity} a(n)/a(n-1) = 1 for n mod 2 = 1. - Branman
In 2001, Brockhaus proved that if the binary Reverse and Add trajectory of an integer contains an integer of one of four specific given  forms, then the trajectory never reaches a palindrome. In the case of 26, that would be 3(2^(2k + 1) - 2^k), with k = 3 corresponding to 360. - Alonso del Arte, Jun 02 2012

			In binary, 26 is 11010.
a(1) = 37 because 11010 + 01011 = 100101, or 37.
a(2) = 78 because 100101 + 101001 = 1001110, or 78.

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

Cf. A035522, A061561, A066059, A077076, A077077.

binRA[n_] := If[Reverse[IntegerDigits[n, 2]] == IntegerDigits[n, 2], n, FromDigits[Reverse[IntegerDigits[n, 2]], 2] + n]; NestList[binRA, 26, 100]

cp's OEIS Frontend

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

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

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A062128 In base 2: 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.

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A075299 Trajectory of 290 under the Reverse and Add! operation carried out in base 4, written in base 10.

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A062129 In base 2: start with n; add to itself with digits reversed; if palindrome, stop; otherwise repeat; a(n) gives palindrome at which it stops, or -1 if no palindrome is ever reached.

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A066450 a(n) is the conjectured value of the minimal number to which repeated application of the "reverse and add!" algorithm in base n does not terminate in a palindrome. If there is no such number in base n, then a(n) := -1.

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A076247 Trajectory of 1059774 under the Reverse and Add! operation carried out in base 4, written in base 10.

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A076248 Trajectory of 1059831 under the Reverse and Add! operation carried out in base 4, written in base 10.

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