A036839 -id:A036839 - 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

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

A056964 a(n) = n + reversal of digits of n.

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 33, 44, 55, 66, 77, 88, 99, 110, 121, 132, 44, 55, 66, 77, 88, 99, 110, 121, 132, 143, 55, 66, 77, 88, 99, 110, 121, 132, 143, 154, 66, 77, 88, 99, 110

Offset: 0

Henry Bottomley, Jul 18 2000

If n has an even number of digits then a(n) is a multiple of 11.

Also called the Reverse and Add!, or RADD operation. Iteration of this function leads to the definition of Lychrel and related numbers, cf. A023108, A063048, A088753, A006960, and many others. - M. F. Hasler, Apr 13 2019

			a(17) = 17 + 71 = 88.

Indranil Ghosh, Table of n, a(n) for n = 0..20000 (first 1001 terms from T. D. Noe)
Eric Weisstein's World of Mathematics, Reverse-Then-Add Sequence
Index entries for sequences related to Reverse and Add!

Cf. A004086, A056965, A067030.
Differs from A052008 when n=101 and a(101)=202 while A052008(101)=121
Cf. A036839.

a056964 n = n + a004086 n  -- Reinhard Zumkeller, Oct 14 2011

Table[n+FromDigits[Reverse[IntegerDigits[n]]],{n,0,100}] (* Harvey P. Dale, Jul 19 2014 *)
#+IntegerReverse[#]&/@Range[0,100] (* Harvey P. Dale, Jul 31 2025 *)

A056964(n)=fromdigits(Vecrev(digits(n)))+n \\ Charles R Greathouse IV, Oct 28 2014

def A056964(n): return n+int(str(n)[::-1]) # Indranil Ghosh, Jan 29 2017

a(n) = n + A004086(n) = 2*n - A056965(n).

n < a(n) < 11n for n > 0. - Charles R Greathouse IV, Nov 17 2022

A009994 Numbers with digits in nondecreasing order.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 22, 23, 24, 25, 26, 27, 28, 29, 33, 34, 35, 36, 37, 38, 39, 44, 45, 46, 47, 48, 49, 55, 56, 57, 58, 59, 66, 67, 68, 69, 77, 78, 79, 88, 89, 99, 111, 112, 113, 114, 115, 116, 117, 118, 119, 122

Offset: 1

N. J. A. Sloane

Record values and occurrences of A004185. - Reinhard Zumkeller, Dec 05 2009
A193581(a(n)) = 0. - Reinhard Zumkeller, Aug 10 2011
This sequence was used by the U.S. Bureau of the Census in the mid-1950s to numerically code the alphabetical list of counties within a state (with some modifications for Texas). The 3-digit code has a "self-policing element" built into it and "was fairly effective in detecting the transposition of 2 digits." (Hanna 1959). - Randy A. Becker, Dec 11 2017

Amarnath Murthy and Robert J. Clarke, Some Properties of Staircase sequence, Mathematics & Informatics Quarterly, Volume 11, No. 4, November 2001.
Frank A. Hanna, The Compilation of Manufacturing Statistics. U.S. Department of Commerce, Bureau of the Census, 1959.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Reinhard Zumkeller)
David Applegate, Marc LeBrun, N. J. A. Sloane, Dismal Arithmetic, J. Int. Seq. 14 (2011) # 11.9.8.
David Radcliffe and Brendan McKay, Re: A009994, SeqFan list, Jul 29 2019
Eric Weisstein's World of Mathematics, Digit.
Index entries for 10-automatic sequences.

Apart from the first term, a subsequence of A052382. A254143 is a subsequence.

Cf. A152054, A036839.

import Data.Set (fromList, deleteFindMin, insert)
a009994 n = a009994_list !! n
a009994_list = 0 : f (fromList [1..9]) where
   f s = m : f (foldl (flip insert) s' $ map (10*m +) [m `mod` 10 ..9])
         where (m,s') = deleteFindMin s
-- Reinhard Zumkeller, Aug 10 2011

[k:k in [0..122]|Sort(Intseq(k)) eq Reverse(Intseq(k))]; // Marius A. Burtea, Jul 28 2019

A[0]:= [0]: A[1]:= [$1..9]:
for d from 2 to 4 do
  A[d]:= map(t -> seq(10*t+i,i=(t mod 10) .. 9), A[d-1]):
od:
seq(op(A[d]),d=0..4); # Robert Israel, Jul 28 2019

Select[Range[0, 125], LessEqual@@IntegerDigits[#] &] (* Ray Chandler, Oct 25 2011 *)

is(n)=n=digits(n);n==vecsort(n) \\ Charles R Greathouse IV, Dec 03 2013

from itertools import combinations_with_replacement
def A009994generator():
    yield 0
    l = 1
    while True:
        for i in combinations_with_replacement('123456789',l):
            yield int(''.join(i))
        l += 1 # Chai Wah Wu, Nov 11 2015

def hasDigitsSorted(n: Int): Boolean = {
  val digSort = Integer.parseInt(n.toString.toCharArray.sorted.mkString)
  n == digSort
}
(0 to 200).filter(hasDigitsSorted()) // _Alonso del Arte, Apr 20 2020

a(n) >> exp(n^(1/10)). - Charles R Greathouse IV, Mar 15 2014

a(n) ~ 10^((9! n)^(1/9) - 5), since 10^(d - 1) <= a(n) < 10^d for binomial(d + 8, 9) < n <= binomial(d + 9, 9) = (d + 5 - epsilon)^9 / 9!. Using epsilon = 10/(3n) + o(1/n) gives more precise estimate. [Following Radcliffe and McKay, cf. SeqFan list.] - M. F. Hasler, Jul 30 2019

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

Original entry on oeis.org

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

A114611 Eventual period of the RATS sequence, where 0 indicates a divergent sequence.

Original entry on oeis.org

0, 0, 8, 0, 0, 8, 0, 0, 2, 0, 0, 8, 0, 0, 8, 0, 0, 2, 0, 0, 8, 0, 0, 8, 0, 0, 2, 0, 18, 8, 0, 0, 8, 0, 0, 2, 0, 18, 8, 0, 0, 8, 0, 0, 2, 0, 18, 8, 18, 0, 8, 0, 0, 2, 0, 18, 8, 18, 0, 8, 0, 0, 2, 0, 18, 8, 18, 0, 2, 0, 0, 2, 0, 18, 8, 18, 0, 2, 0, 0, 2, 0, 18, 8, 18, 0, 2, 0, 0, 2, 0, 18, 8, 18, 0, 2

Offset: 1

Eric W. Weisstein, Dec 16 2005

a(A001651(n)) = 0; a(A114612(n)) = 2; a(A114613(n)) = 3; a(A114614(n)) = 8; a(A114615(n)) = 14; a(A114616(n)) = 18. - Reinhard Zumkeller, Mar 14 2012

Row 10 of A288537. - Andrey Zabolotskiy, Jun 14 2017

Eric Weisstein's World of Mathematics, RATS Sequence

Cf. A036839, A161593, A288536, A288537.

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

A066710 RATS: Reverse Add Then Sort the digits applied to previous term, starting with 3.

Original entry on oeis.org

3, 6, 12, 33, 66, 123, 444, 888, 1677, 3489, 12333, 44556, 111, 222, 444, 888, 1677, 3489, 12333, 44556, 111, 222, 444, 888, 1677, 3489, 12333, 44556, 111, 222, 444, 888, 1677, 3489, 12333, 44556, 111, 222, 444, 888, 1677, 3489, 12333

Offset: 1

N. J. A. Sloane, Jan 19 2002

a(1) = A114614(1) = 3; A114611(3) = 8. [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.
Eric Weisstein's World of Mathematics, RATS Sequence
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,1).

Cf. A004000, A036839, A066711, A209878, A209879, A209880.

a066710_list = iterate a036839 3  -- Reinhard Zumkeller, Mar 14 2012

f[k_] := Module[{m = FromDigits[Reverse[IntegerDigits[k]]]}, FromDigits[ Sort[ IntegerDigits[k + m]]]]; NestList[f, 3, 50] (* Harvey P. Dale, Jan 18 2011 *)

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 8.
a(n+1) = A036839(a(n)). [Reinhard Zumkeller, Mar 14 2012]
From Chai Wah Wu, Feb 07 2020: (Start)
a(n) = a(n-8) for n > 14.
G.f.: x*(-99*x^13 - 45*x^12 - 44523*x^11 - 12321*x^10 - 3483*x^9 - 1674*x^8 - 888*x^7 - 444*x^6 - 123*x^5 - 66*x^4 - 33*x^3 - 12*x^2 - 6*x - 3)/(x^8 - 1). (End)

A114612 Starting numbers for which the RATS sequence has eventual period 2.

Original entry on oeis.org

9, 18, 27, 36, 45, 54, 63, 69, 72, 78, 81, 87, 90, 96, 99, 108, 114, 117, 120, 126, 135, 144, 153, 156, 162, 171, 180, 189, 198, 207, 213, 216, 225, 234, 243, 252, 255, 261, 270, 279, 288, 297, 306, 312, 315, 324, 333, 342, 351, 354, 360, 369, 378, 387, 396

Offset: 1

Eric W. Weisstein, Dec 16 2005

A114611(a(n)) = 2. - Reinhard Zumkeller, Mar 14 2012

Eric Weisstein's World of Mathematics, RATS Sequence

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

Cf. A066711, A036839.

A114613 Starting numbers for which the RATS sequence has eventual period 3.

Original entry on oeis.org

20169, 20709, 21159, 22149, 23139, 24129, 25119, 26109, 27099, 28089, 29079, 30159, 30168, 30708, 30789, 31149, 31158, 31779, 32139, 32148, 32769, 33129, 33138, 33759, 34119, 34128, 34749, 35109, 35118, 35739, 36108, 36729, 37098

Offset: 1

Eric W. Weisstein, Dec 16 2005

A114611(a(n)) = 3. - Reinhard Zumkeller, Mar 14 2012

Eric Weisstein's World of Mathematics, RATS Sequence

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

Cf. A209878, A036839.

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

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A056964 a(n) = n + reversal of digits of n.

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A009994 Numbers with digits in nondecreasing order.

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

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A114611 Eventual period of the RATS sequence, where 0 indicates a divergent sequence.

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

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