A005658 If n appears so do 2n, 3n+2, 6n+3.

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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 and 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

Crossrefs

Cf. A002977, A185661.

Programs

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

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 *)

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

Extensions

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