A066710 -id:A066710 - cp's OEIS frontend

A004000 RATS: Reverse Add Then Sort the digits applied to previous term, starting with 1.

1, 2, 4, 8, 16, 77, 145, 668, 1345, 6677, 13444, 55778, 133345, 666677, 1333444, 5567777, 12333445, 66666677, 133333444, 556667777, 1233334444, 5566667777, 12333334444, 55666667777, 123333334444, 556666667777, 1233333334444, 5566666667777, 12333333334444

Offset: 1

N. J. A. Sloane

It is conjectured that no matter what the starting term is, repeatedly applying RATS leads either to this sequence or into a cycle of finite length, such as those in A066710 and A066711.

			668 -> 668 + 866 = 1534 -> 1345.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..2002 (first 200 terms from T. D. Noe)
R. K. Guy, Conway's RATS and other reversals, Amer. Math. Monthly, 96 (1989), 425-428.
Eric Weisstein's World of Mathematics, RATS Sequence.

Cf. A036839, A066710, A066711, A066713, A164338, A161593, A114611, A114612, A209878, A209879, A209880.

a004000_list = iterate a036839 1  -- Reinhard Zumkeller, Mar 14 2012

[ n eq 1 select 1 else Seqint(Reverse(Sort(Intseq(p + Seqint(Reverse(Intseq(p))) where p is Self(n-1))))) : n in [1..10]]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 20061

read transforms; RATS := n -> digsort(n + digrev(n)); b := [1]; t := [1]; for n from 1 to 50 do t := RATS(t); b := [op(b),t]; od: b;

NestList[FromDigits[Sort[IntegerDigits[#+FromDigits[Reverse[ IntegerDigits[#]]]]]]&,1,30] (* Harvey P. Dale, Nov 29 2011 *)

step(n)=fromdigits(vecsort(digits(n+fromdigits(Vecrev(digits(n)))))) \\ Charles R Greathouse IV, Jun 23 2017

l = [0, 1]
for n in range(2, 51):
    x = str(l[n - 1])
    l.append(int(''.join(sorted(str(int(x) + int(x[::-1]))))))
print(l[1:]) # Indranil Ghosh, Jul 05 2017

Let a(n) = k, form m by Reversing the digits of k, Add m to k Then Sort the digits of the sum into increasing order to get a(n+1).
a(n+1) = A036839(a(n)). - Reinhard Zumkeller, Mar 14 2012
A010888(a(n)) = A153130(n-1). - Ivan N. Ianakiev, Nov 27 2014
a(2n-1) = (37 * 10^(n-3) + 3332)/3, n >= 11; a(2n) = (167 * 10^(n-3) + 3331)/3, n >= 10. - Jianing Song, May 06 2021

Entry revised by N. J. A. Sloane, Jan 19 2002

A066711 RATS: Reverse Add Then Sort the digits applied to previous term, starting with 9.

Original entry on oeis.org

9, 18, 99, 189, 117, 288, 117, 288, 117, 288, 117, 288, 117, 288, 117, 288, 117, 288, 117, 288, 117, 288, 117, 288, 117, 288, 117, 288, 117, 288, 117, 288, 117, 288, 117, 288, 117, 288, 117, 288, 117, 288, 117, 288, 117, 288, 117, 288, 117

Offset: 1

N. J. A. Sloane, Jan 19 2002

a(1) = A114612(1) = 9; A114611(3) = 2. - Reinhard Zumkeller, Mar 14 2012

			668 -> 668 + 866 = 1534 -> 1345.

R. K. Guy, Conway's RATS and other reversals, Amer. Math. Monthly, 96 (1989), 425-428.
J. Thiel, Conway’s RATS Sequences in Base 3, Journal of Integer Sequences, 15 (2012), #12.9.2. - _N. J. A. Sloane_, Jan 02 2013
Eric Weisstein's World of Mathematics, RATS Sequence
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A004000, A036839, A066710, A033896, A114611, A114612.

Cf. A004000, A066710, A209878, A209879, A209880.

a066711_list = iterate a036839 9  -- Reinhard Zumkeller, Mar 14 2012

NestList[ FromDigits[ Sort[ IntegerDigits[# + FromDigits[ Reverse[ IntegerDigits[#]]]]]] &, 9, 48] (* Jayanta Basu, Aug 13 2013 *)
Join[{9, 18, 99, 189},LinearRecurrence[{0, 1},{117, 288},45]] (* Ray Chandler, Aug 25 2015 *)

from itertools import accumulate
def rats(anm1, _):
    return int("".join(sorted(str(anm1 + int(str(anm1)[::-1])))))
print(list(accumulate([9]*49, rats))) # Michael S. Branicky, Sep 18 2021

Let a(n) = k, form m by Reversing the digits of k, Add m to k Then Sort the digits of the sum into increasing order to get a(n+1).
Periodic with period 2.
a(n+1) = A036839(a(n)). - Reinhard Zumkeller, Mar 14 2012
G.f.: x*(-99*x^5 - 18*x^4 - 171*x^3 - 90*x^2 - 18*x - 9)/(x^2 - 1). - Chai Wah Wu, Feb 07 2020

A114614 Starting numbers for which the RATS sequence has eventual period 8.

Original entry on oeis.org

3, 6, 12, 15, 21, 24, 30, 33, 39, 42, 48, 51, 57, 60, 66, 75, 84, 93, 102, 105, 111, 123, 129, 132, 138, 141, 147, 150, 159, 165, 168, 174, 177, 183, 186, 192, 195, 201, 204, 210, 219, 222, 228, 231, 237, 240, 246, 249, 258, 264, 267, 273, 276, 282, 285, 291

Offset: 1

Eric W. Weisstein, Dec 16 2005

A114611(a(n)) = 8. - Reinhard Zumkeller, Mar 14 2012

Eric Weisstein's World of Mathematics, RATS Sequence

Cf. A114611, A114612, A114613, A114614, A114615, A114616.

Cf. A066710, A036839.

A161593 Lengths of new periods in the RATS sequence (0 replacing infinity).

Original entry on oeis.org

0, 8, 2, 18, 2, 2, 2, 14, 2, 3, 2, 2, 2, 6

Offset: 1

J. H. Conway and Tanya Khovanova, Jun 14 2009

The values A114611(j) for those starting values j of the RATS mapping x->A036839(x) which end in cycles that cannot be reached starting from any smaller j.
Every integer > 1 appears in this sequence. - Andrey Zabolotskiy, Jun 11 2017
For other terms see Branicky link. - Michael S. Branicky, Dec 30 2022

			a(1)=A114611(0). a(2)=A114611(j=3)=8 with a cycle of length 8 shown in A066710.
A114611(j=6)=8 does not contribute because the cycle is the same as reached from j=3.
a(3)=A114611(9)=2 with a new cycle of length 2 shown in A066711.
A114611(j=12)=8 does not contribute because the cycle is the same as reached from j=3.
A114611(j=15)=8 does not contribute because 15->66->123 is the cycle as reached from j=3.
A114611(j=18)=2 does not contribute because the cycle is the same as reached from j=9.
A114611(j=21)=8 does not contribute because 21->33->66 reaches the same cycle as started from j=3.
a(4)=A114611(j=29)=18.

Michael S. Branicky, RATS Sequence Cycles.
Curtis Cooper, RATS.
Curtis Cooper and Robert E. Kennedy, Base 10 RATS Cycles and Arbitrarily Long Base 10 RATS Cycles, Applications of Fibonacci numbers, Vol. 8, Kluwer Acad. Publ., Dordrecht, 1999, pages 83-93.
Tanya Khovanova, Destinies of Numbers.

Cf. A004000, A036839, A066711, A066710, A288537.

Cf. A161590, A161592, A161596.

Comment and examples added by R. J. Mathar, Jul 07 2009

a(9)-a(14) from Michael S. Branicky, Dec 30 2022

A209878 RATS: Reverse Add Then Sort the digits applied to previous term, starting with 20169.

Original entry on oeis.org

20169, 111267, 337788, 1122255, 4446666, 1111113, 2222244, 4446666, 1111113, 2222244, 4446666, 1111113, 2222244, 4446666, 1111113, 2222244, 4446666, 1111113, 2222244, 4446666, 1111113, 2222244, 4446666, 1111113, 2222244, 4446666, 1111113, 2222244, 4446666

Offset: 1

Reinhard Zumkeller, Mar 14 2012

A114613(1) = 20169 is the smallest starting number for a RATS trajectory leading to a cycle of length 3: A114611(20169) = 3;

a(n + 3) = a(n) for n > 4.

Eric Weisstein's World of Mathematics, RATS Sequence
Index entries for linear recurrences with constant coefficients, signature (0, 0, 1).

Cf. A004000, A066710, A066711, A209879, A209880.

a209878 n = a209878_list !! (n-1)
a209878_list = iterate a036839 20169

Join[{20169, 111267, 337788, 1122255},LinearRecurrence[{0, 0, 1},{4446666, 1111113, 2222244},25]] (* Ray Chandler, Aug 25 2015 *)

a(n + 1) = A036839(a(n)).

A209879 RATS: Reverse Add Then Sort the digits applied to previous term, starting with 6999.

Original entry on oeis.org

6999, 15699, 11355, 66666, 123333, 445566, 111111, 222222, 444444, 888888, 1677777, 3455589, 11112333, 33444444, 77778888, 156666666, 123333378, 666669999, 1356666666, 123333789, 11111667, 22777788, 11115555, 66666666, 123333333, 445566666, 111122223

Offset: 1

Reinhard Zumkeller, Mar 14 2012

A114615(1) = 6999 is the smallest starting number for a RATS trajectory leading to a cycle of length 14: A114611(6999) = 14;

a(n + 14) = a(n) for n > 25.

Reinhard Zumkeller, Table of n, a(n) for n = 1..100
Eric Weisstein's World of Mathematics, RATS Sequence
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A004000, A066710, A209878, A066711, A209880.

a209879 n = a209879_list !! (n-1)
a209879_list = iterate a036839 6999

rats[n_]:=Module[{idnr=FromDigits[Reverse[IntegerDigits[n]]]}, FromDigits[ Sort[ IntegerDigits[idnr+n]]]]; NestList[rats,6999,30] (* Harvey P. Dale, May 29 2014 *)

a(n + 1) = A036839(a(n)).

A209880 RATS: Reverse Add Then Sort the digits applied to previous term, starting with 29.

Original entry on oeis.org

29, 112, 233, 556, 1112, 2233, 5555, 1111, 2222, 4444, 8888, 16777, 34589, 112333, 444455, 889999, 1788899, 1177777, 4558889, 13444447, 77888888, 156667777, 233444489, 1112278888, 11999, 11119, 1223, 4444, 8888, 16777, 34589, 112333, 444455, 889999, 1788899

Offset: 1

Reinhard Zumkeller, Mar 14 2012

A114616(1) = 29 is the smallest starting number for a RATS trajectory leading to a cycle of length 18: A114611(29) = 18;

a(n + 18) = a(n) for n > 9.

Eric Weisstein's World of Mathematics, RATS Sequence
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A004000, A066710, A209878, A066711, A209879.

a209880 n = a209880_list !! (n-1)
a209880_list = iterate a036839 29

NestList[FromDigits[Sort[IntegerDigits[#+IntegerReverse[#]]]]&,29,40] (* or *) PadRight[{29,112,233,556,1112,2233,5555,1111,2222},50,{4558889,13444447,77888888,156667777,233444489,1112278888,11999,11119,1223,4444,8888,16777,34589,112333,444455,889999,1788899,1177777}] (* Harvey P. Dale, Sep 17 2018 *)

a(n + 1) = A036839(a(n)).

A161596 Numbers in cycles of RATS sequences.

Original entry on oeis.org

78, 111, 117, 156, 222, 288, 444, 888, 1223, 1677, 3489, 4444, 8888, 11119, 11127, 11667, 11999, 12333, 16777, 23388, 27888, 34589, 44556, 111177, 112333, 228888, 444455, 889999, 1111113, 1177777, 1788899, 2222244, 4446666, 4558889, 11144445, 13444447, 55556688

Offset: 1

J. H. Conway and Tanya Khovanova, Jun 14 2009

The set of all numbers in any cycle of RATS sequences, sorted into natural order.
This implies that for any value a(j) in this sequence, A036839(a(j)) is again member of the sequence.
See Branicky link for larger terms. - Michael S. Branicky, Dec 30 2022

			The numbers 111, 222, 444, 888, 1677, 3489, 12333 and 44556 are in the sequence because they are in the cycle shown in A066710. The numbers 117 and 288 are in the cycle demonstrated in A066711.
The numbers 4444, 8888, 16777, 34589, 112333, 444455, ..., 1112278888, 11999, 1119, 1223 are in the cycle started at A161590(4). The numbers 11127 and 23388 are in the cycle started at A161590(7).

Michael S. Branicky, RATS Sequence Cycles.

Cf. A004000, A036839, A066711, A066710, A161590, A161592.

Descriptive comment and examples added by R. J. Mathar, Jul 08 2009

a(20) and beyond from Michael S. Branicky, Dec 30 2022

A161590 Initial value x of a RATS trajectory x->A036839(x) ending in a cycle unreachable by any smaller initial value.

Original entry on oeis.org

1, 3, 9, 29, 69, 2079, 3999, 6999, 10677, 20169, 10049598, 20008989, 100014888, 100074268

Offset: 1

J. H. Conway and Tanya Khovanova, Jun 14 2009, Jul 04 2009

This is one way of book-keeping of new "destinies" (the smallest element of the cycle that the trajectory ends up in).
The value 1 is a placeholder for all non-cyclic trajectories.
Next terms are respectively <= 10000122228, 20000666679, 2000001113379, 2000001113559, 9999999999999, 100000044444447.  See Branicky link for further upper bounds. - Michael S. Branicky, Dec 30 2022

			The RATS (Reverse Add Then Sort) algorithm applied to 69 produces a sequence 69, 156, 78, 156, 78, ...
Its cycle {156, 78} appears not if the algorithm is started with any number in the range 0 to 68, so 69 is added to the sequence.

Michael S. Branicky, RATS Sequence Cycles.
Tanya Khovanova, Destinies of Numbers. [From _Tanya Khovanova_, Aug 10 2009]

Cf. A036839, A066710, A066711, A004000, A114611, A161592, A161593, A161596.

10677, 20169 from Wouter Meeussen, Jul 04 2009
Definition rephrased by R. J. Mathar, Jul 08 2009
a(11)-a(14) from Michael S. Branicky, Dec 30 2022

A161592 Except for the first term the number in the sequence is the smallest number in a new cycle of a RATS sequence with a new destiny. The first term is the best analog of this for the "infinite cycle".

Original entry on oeis.org

12334444, 111, 117, 1223, 78, 111177, 11127, 11144445, 11667, 1111113

Offset: 1

J. H. Conway & Tanya Khovanova, Jun 14 2009

"Destiny" means the smallest element of the cycle that the trajectory ends up in.

All seeds except those generating the cycles listed here produce an open non-cyclic family (thus without lowest element) but with a regular structure like 12334444, 55667777, 123334444, 556667777, 1233334444, 5566667777,..., and with an arbitrary start-up like 1, 2, 4, 8, 16, 77, 145, 668, 1345, 6677, 13444, 55778, 133345, 666677, 1333444, 5567777, 12333445, 66666677, 133333444, 556667777, 1233334444, 5566667777, 12333334444, 55666667777, 123333334444, ... Notice that here we fall into the regular regime starting with 1233334444 (four threes). The sequence gives 12-(two threes)-4444 as a representative with index 1. - Wouter Meeussen, Jul 26 2009

Tanya Khovanova, Destinies of Numbers - _Tanya Khovanova_, Aug 10 2009

Cf. A004000, A036839, A066710, A066711, A161590.

Cf. A161593, A161596. - Tanya Khovanova, Aug 10 2009

11667, 1111113 from Wouter Meeussen, Jul 04 2009

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A004000 RATS: Reverse Add Then Sort the digits applied to previous term, starting with 1.

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A066711 RATS: Reverse Add Then Sort the digits applied to previous term, starting with 9.

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A114614 Starting numbers for which the RATS sequence has eventual period 8.

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A161593 Lengths of new periods in the RATS sequence (0 replacing infinity).

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A209878 RATS: Reverse Add Then Sort the digits applied to previous term, starting with 20169.

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A209879 RATS: Reverse Add Then Sort the digits applied to previous term, starting with 6999.

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A209880 RATS: Reverse Add Then Sort the digits applied to previous term, starting with 29.

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