A004000 -id:A004000 - cp's OEIS frontend

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

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

A288537 Array A(b,n) by upward antidiagonals (b>1, n>0): the eventual period of the RATS sequence in base b starting from n; 0 is for infinity.

Original entry on oeis.org

1, 3, 1, 2, 3, 1, 2, 2, 3, 1, 8, 2, 2, 3, 1, 4, 8, 2, 2, 3, 1, 3, 4, 8, 2, 2, 3, 1, 2, 3, 2, 8, 2, 2, 3, 1, 0, 2, 3, 4, 2, 2, 2, 3, 1, 28, 0, 2, 3, 4, 8, 2, 2, 3, 1, 90, 28, 8, 2, 6, 2, 8, 2, 2, 3, 1, 8, 90, 28, 0, 2, 3, 4, 8, 2, 2, 3, 1, 72, 8, 90, 28, 0, 2

Offset: 2

Andrey Zabolotskiy, Jun 11 2017

Eventual period of n under the mapping x->A288535(b,x), or 0 if there is a divergence and thus no eventual period.
For b = 3*2^m - 2 with m>1, row b contains all sufficiently large even integers if m is odd, or just all sufficiently large integers if m is even.
For b = 1 or 10 (mod 18) or b = 1 (mod (2^q-1)^2) with q>2, there are 0's in row b.
Conway conjectured that in row (base) 10, all 0's correspond to the same divergent RATS sequence called the Creeper (A164338). In Thiel's terms, it is quasiperiodic with quasiperiod 2, i.e., after every 2 steps the number of one of the digits (in this case, 3 or 6) increases by 1 while other digits stay unchanged. In other bases, 0's may correspond to different divergent RATS sequences. Thiel conjectured that the divergent RATS sequences are always quasiperiodic.

			In base 3, the RATS mapping acts as 1 -> 2 -> 4 (11 in base 3) -> 8 (22 in base 3) -> 13 (112 in base 3) -> 4, which has already been seen 3 steps ago, so A(3,1)=3.
The array begins:
1, 1, 1, 1, 1, 1, ...
3, 3, 3, 3, 3, 3, ...
2, 2, 2, 2, 2, 2, ...
2, 2, 2, 2, 2, 2, ...
8, 8, 8, 8, 2, 8, ...
4, 4, 2, 4, 4, 2, ...
3, 3, 3, 3, 6, 3, ...
2, 2, 2, 2, 2, 2, ...
0, 0, 8, 0, 0, 8, ...
28, 28, 28, 28, 2, 28, ...
90, 90, 90, 90, 90, 90 ...

Curtis Cooper, RATS.
R. K. Guy, Conway's RATS and other reversals, Amer. Math. Monthly, 96 (1989), 425-428.
S. Shattuck and C. Cooper, Divergent RATS sequences, Fibonacci Quart., 39 (2001), 101-106.
J. Thiel, Conway’s RATS Sequences in Base 3, Journal of Integer Sequences, 15 (2012), Article 12.9.2.
J. Thiel, On RATS sequences in general bases, Integers, 14 (2014), #A50.
Eric Weisstein's World of Mathematics, RATS Sequence.
Index entries for sequences related to RATS: Reverse, Add Then Sort

Cf. A004000, A036839, A114611 (row 10), A161593, A288535, A288536 (column 1).

A(2^t,1)=t.

A(3,3^A134067(p)-1)=p+3.

A288535 Array RATS(b,n) by upward antidiagonals: Reverse Add Then Sort digits of n>0 in base b>1.

Original entry on oeis.org

1, 2, 3, 2, 4, 3, 2, 1, 4, 3, 2, 4, 6, 8, 3, 2, 4, 6, 5, 4, 3, 2, 4, 1, 8, 10, 8, 7, 2, 4, 6, 8, 6, 15, 4, 3, 2, 4, 6, 8, 10, 12, 5, 14, 3, 2, 4, 6, 1, 10, 7, 18, 10, 4, 15, 2, 4, 6, 8, 10, 12, 14, 24, 15, 8, 3, 2, 4, 6, 8, 10, 12, 8, 21, 6, 5, 4, 15, 2, 4, 6

Offset: 2

Andrey Zabolotskiy, Jun 11 2017

			17 in base 3 is 122, 122+221=1120->112 which is 14 in decimal, thus RATS(3,17)=14.
The array begins:
1, 3, 3, 3,  3,  3,  7, ...
2, 4, 4, 8,  4,  8,  4, ...
2, 1, 6, 5, 10, 15,  5, ...
2, 4, 6, 8,  6, 12, 18, ...
2, 4, 1, 8, 10,  7, 14, ...

J. Thiel, On RATS sequences in general bases, Integers, 14 (2014), #A50.
Index entries for sequences related to RATS: Reverse, Add Then Sort

Cf. A004000, A036839 (row 10), A191780, A288536, A288537.

rats[n_, b_: 10] := FromDigits[Sort[IntegerDigits[n + FromDigits[Reverse[IntegerDigits[n, b]], b], b]], b];
Flatten[Table[rats[n, s + 2 - n], {s, 20}, {n, s}]]

A288536 The eventual period of the RATS sequence in base n starting from 1; 0 is for infinity.

Original entry on oeis.org

1, 3, 2, 2, 8, 4, 3, 2, 0, 28, 90, 8, 72, 3, 4, 2, 64, 0, 18, 4, 18, 20, 396, 8, 160, 120, 18, 6, 28, 4, 5, 2, 210, 384, 240, 0, 648, 1242, 240, 4, 660, 18, 798, 380, 852, 1298, 1771, 8, 0, 160, 16, 372, 520, 1404, 1740, 6, 36, 2072, 1856, 380, 300, 215, 6, 2, 3384, 50, 2310, 3784, 2904

Offset: 2

Andrey Zabolotskiy, Jun 11 2017

Eventual period of 1 under the mapping x->A288535(n,x), or 0 if there is a divergence and thus no eventual period.
Column 1 of A288537.
In Thiel's terms, the zeroes a(10), a(19), and a(37) correspond to quasiperiodic divergent RATS sequences with quasiperiod 2, while a(50)=0 corresponds to a sequence with quasiperiod 3.

			In base 3, the RATS mapping acts as 1 -> 2 -> 4 (11 in base 3) -> 8 (22 in base 3) -> 13 (112 in base 3) -> 4, which has already been seen 3 steps ago, so a(3)=3.

S. Shattuck and C. Cooper, Divergent RATS sequences, Fibonacci Quart., 39 (2001), 101-106.
J. Thiel, On RATS sequences in general bases, Integers, 14 (2014), #A50.
Index entries for sequences related to RATS: Reverse, Add Then Sort

Cf. A004000, A114611, A288535, A288537, A237671, A072137.

A066713 RATS(2^n): Reverse Add the digits of 2^n, Then Sort: a(n) = A036839(2^n).

Original entry on oeis.org

2, 4, 8, 16, 77, 55, 11, 499, 89, 277, 2255, 145, 11, 1111, 44567, 111499, 12299, 1234, 3467, 113467, 677789, 144556, 1222889, 14445667, 4577789, 55669999, 1134899, 11356999, 12237899, 445557799, 1223555555, 11113366, 1122222266

Offset: 0

N. J. A. Sloane, Jan 19 2002

A114611(a(n)) = 0, as A114611(A000079(n)) = 0. - Reinhard Zumkeller, Mar 14 2012

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, RATS Sequence

See A004000, A036839 for more information.

a066713 = a036839 . (2 ^)  -- Reinhard Zumkeller, Mar 14 2012

def A066713(n):
    m = 2**n
    return int(''.join(sorted(str(m+int(str(m)[::-1]))))) # Chai Wah Wu, Feb 07 2020

A079320 CATS sequence: cube-add-then-sort variation of RATS (reverse, add then sort) sequence.

Original entry on oeis.org

1, 3, 68, 13, 222, 35, 25, 378, 11, 1234, 147, 122, 2578, 339, 1124, 349, 558, 6788, 28, 2289, 167, 1129, 13488, 1556, 1267, 1179, 1289, 12448, 237, 22289, 238, 3579, 33389, 1249, 24669, 569, 1459, 35589, 446, 26689, 1347, 5579, 22588, 1179, 23789, 1378

Offset: 2

Michael Joseph Halm, Feb 13 2003

			a(8)= 25 because 8^3 + 8 = 512 + 8 = 520, sort(520) = 25.

Harvey P. Dale and Indranil Ghosh, Table of n, a(n) for n = 2..20000 (terms 2..1000 from Harvey P. Dale)
M. J. Halm, Sequences

Cf. A004000.

Array[FromDigits[Sort[IntegerDigits[#^3+#]]]&,50,2] (* Harvey P. Dale, Feb 28 2013 *)

def A079320(n):
    x=str(n+n**3)
    return int("".join(sorted(x))) # Indranil Ghosh, Jan 29 2017

a(n) = sort_digits(n^3 + n).

A333302 Numbers produced by iteratively sorting the digits of the last number from largest to smallest in base 10 and then doubling, starting with the number 1.

Original entry on oeis.org

1, 2, 4, 8, 16, 122, 442, 884, 1768, 17522, 150442, 1088420, 17684200, 175284200, 1750844200, 17508842000, 177508420000, 1755084200000, 17510842000000, 175084220000000, 1750844200000000, 17508842000000000, 177508420000000000, 1755084200000000000

Offset: 1

Ethan Hulinsky, Mar 14 2020

It appears that the first 8 digits begin to cycle in a period of 6 while adding one zero every iteration.

Cf. A004000, A057615, A095197.

a[1] = 1; a[n_] := a[n] = 2 FromDigits@ Reverse@ Sort@ IntegerDigits@ a[n-1]; Array[a, 24] (* Giovanni Resta, Apr 15 2020 *)

lista(nn) = {my(a=1); print1(a, ", "); for (n=2, nn, a = 2*fromdigits(vecsort(digits(a),,4)); print1(a, ", "););} \\ Michel Marcus, Apr 16 2020

cp's OEIS Frontend

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

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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".

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A288537 Array A(b,n) by upward antidiagonals (b>1, n>0): the eventual period of the RATS sequence in base b starting from n; 0 is for infinity.

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Formula

A288535 Array RATS(b,n) by upward antidiagonals: Reverse Add Then Sort digits of n>0 in base b>1.

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Mathematica

A288536 The eventual period of the RATS sequence in base n starting from 1; 0 is for infinity.

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Examples

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A066713 RATS(2^n): Reverse Add the digits of 2^n, Then Sort: a(n) = A036839(2^n).

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Haskell

Python

A079320 CATS sequence: cube-add-then-sort variation of RATS (reverse, add then sort) sequence.

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Mathematica

Python

Formula

A333302 Numbers produced by iteratively sorting the digits of the last number from largest to smallest in base 10 and then doubling, starting with the number 1.

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Mathematica

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