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

Author

N. J. A. Sloane

Keywords

• base
• nonn
• nice
• easy

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.

Examples

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

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.

Crossrefs

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

Programs

Haskell

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

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

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 *)

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

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

Extensions

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