A004000 RATS: Reverse Add Then Sort the digits applied to previous term, starting with 1. (Formerly M1137)

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

1,2

Comments

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.

References

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

Links

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.

Formula

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

Example

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

Maple

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;

Mathematica

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

Prog

(Magma) [ 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
(Haskell)
a004000_list = iterate a036839 1  -- Reinhard Zumkeller, Mar 14 2012
(PARI) step(n)=fromdigits(vecsort(digits(n+fromdigits(Vecrev(digits(n)))))) \\ Charles R Greathouse IV, Jun 23 2017
(Python)
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

Crossrefs

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

Sequence in context: A012997 A013184 A066713 * A374732 A051300 A001127

Adjacent sequences: A003997 A003998 A003999 * A004001 A004002 A004003

Keyword

base,nonn,nice,easy

Author

N. J. A. Sloane

Extensions

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

Status

approved