A005658 If n appears so do 2n, 3n+2, 6n+3. (Formerly M0969)

1, 2, 4, 5, 8, 9, 10, 14, 15, 16, 17, 18, 20, 26, 27, 28, 29, 30, 32, 33, 34, 36, 40, 44, 47, 50, 51, 52, 53, 54, 56, 57, 58, 60, 62, 63, 64, 66, 68, 72, 80, 83, 86, 87, 88, 89, 92, 93, 94, 98, 99, 100, 101, 102, 104, 105, 106, 108, 110, 111, 112, 114, 116, 120, 122, 123

Offset

1,2

Comments

David Klarner and coauthors studied several sequences of this type. Some of the references here apply generally to this class of sequences.

References

Guy, R. K., Klarner-Rado Sequences. Section E36 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 237, 1994.

J. C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, Amer. Math. Soc., 2010. See pp. 6, 280.

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

Links

R. J. Mathar, Table of n, a(n) for n = 1..15889

R. K. Guy, Letter to N. J. A. Sloane with attachment, 1982

R. K. Guy, Don't try to solve these problems, Amer. Math. Monthly, 90 (1983), 35-41.

Dean G. Hoffman and David A. Klarner, Sets of integers closed under affine operators-the closure of finite sets, Pacific J. Math. 78 (1978), no. 2, 337-344.

Dean G. Hoffman and David A. Klarner, Sets of integers closed under affine operators-the finite basis theorem, Pacific J. Math. 83 (1979), no. 1, 135-144.

David A. Klarner, m-Recognizability of sets closed under certain affine functions, Discrete Appl. Math. 21 (1988), no. 3, 207-214.

David A. Klarner, Karel Post, Some fascinating integer sequences, A collection of contributions in honour of Jack van Lint, Discrete Math. 106/107 (1992), 303-309.

David A. Klarner and R. Rado, Arithmetic properties of certain recursively defined sets, Pacific J. Math. 53 (1974), 445-463.

Eric Weisstein's World of Mathematics, Klarner-Rado Sequence.

Index entries for sequences related to 3x+1 (or Collatz) problem

Maple

ina:= proc(n) evalb(n=1) end:
a:= proc(n) option remember; local k, t;
      if n=1 then 1
    else for k from a(n-1)+1 while not
           (irem(k, 2, 't')=0 and ina(t) or
            irem(k, 3, 't')=2 and ina(t) or
            irem(k, 6, 't')=3 and ina(t) )
         do od: ina(k):= true; k
      fi
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Mar 16 2011

Mathematica

s={1}; Do[a=s[[n]]; s=Union[s, {2a, 3a+2, 6a+3}], {n, 1000}]; s (* Zak Seidov, Mar 15 2011 *)
nxt[n_]:=Flatten[{#, 2#, 3#+2, 6#+3}&/@n]; Take[Union[Nest[nxt, {1}, 5]], 100] (* Harvey P. Dale, Feb 06 2015 *)

Prog

(C++)
#include <stdio.h>
#include <iostream>
#include <set>
using namespace std ;
int main(int argc, char *argv[])
{ const int anmax= 40000 ; set<int> a ; a.insert(1) ; for(int i=0; i< anmax ; i++) { if( a.count(i) ) { if( 2*i<=anmax) a.insert(2*i) ; if( 3*i+2 <= anmax) a.insert(3*i+2) ; if( 6*i+3 <= anmax) a.insert(6*i+3) ; } } int n=1 ; for(int i=0; i < anmax; i++) { if( a.count(i) ) { cout << n << " " << i << endl ; n++ ; } } return 0 ; }
- R. J. Mathar, Aug 20 2006
(Haskell)
import Data.Set (Set, fromList, insert, deleteFindMin)
a005658 n = a005658_list !! (n-1)
a005658_list = klarner $ fromList [1, 2] where
   klarner :: Set Integer -> [Integer]
   klarner s = m : (klarner $
                    insert (2*m) $ insert (3*m+2) $ insert (6*m+3) s')
      where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Mar 14 2011
(PARI) is(n)=if(n<3, return(n>0)); my(k=n%6); if(k==3, return(is(n\6))); if(k==1, return(0)); if(k==5, return(is(n\3))); if(k!=2, return(is(n/2))); is(n\3) || is(n/2) \\ Charles R Greathouse IV, Sep 15 2015

Crossrefs

Cf. A002977, A185661.

Sequence in context: A353386 A101185 A045702 * A166021 A339906 A375906

Adjacent sequences: A005655 A005656 A005657 * A005659 A005660 A005661

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Oct 16 2000

Status

approved