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%I M0969 #46 Jun 07 2019 22:01:08
%S 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,
%T 50,51,52,53,54,56,57,58,60,62,63,64,66,68,72,80,83,86,87,88,89,92,93,
%U 94,98,99,100,101,102,104,105,106,108,110,111,112,114,116,120,122,123
%N If n appears so do 2n, 3n+2, 6n+3.
%C David Klarner and coauthors studied several sequences of this type. Some of the references here apply generally to this class of sequences.
%D Guy, R. K., Klarner-Rado Sequences. Section E36 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 237, 1994.
%D J. C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, Amer. Math. Soc., 2010. See pp. 6, 280.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H R. J. Mathar, <a href="/A005658/b005658.txt">Table of n, a(n) for n = 1..15889</a>
%H R. K. Guy, <a href="/A005658/a005658.pdf">Letter to N. J. A. Sloane with attachment, 1982</a>
%H R. K. Guy, <a href="https://www.jstor.org/stable/2975688">Don't try to solve these problems</a>, Amer. Math. Monthly, 90 (1983), 35-41.
%H Dean G. Hoffman and David A. Klarner, <a href="http://dx.doi.org/10.2140/pjm.1978.78.337">Sets of integers closed under affine operators-the closure of finite sets</a>, Pacific J. Math. 78 (1978), no. 2, 337-344.
%H Dean G. Hoffman and David A. Klarner, <a href="http://dx.doi.org/10.2140/pjm.1979.83.135">Sets of integers closed under affine operators-the finite basis theorem</a>, Pacific J. Math. 83 (1979), no. 1, 135-144.
%H David A. Klarner, <a href="https://doi.org/10.1016/0166-218X(88)90067-4">m-Recognizability of sets closed under certain affine functions</a>, Discrete Appl. Math. 21 (1988), no. 3, 207-214.
%H David A. Klarner, Karel Post, <a href="https://doi.org/10.1016/0012-365X(92)90558-W">Some fascinating integer sequences</a>, A collection of contributions in honour of Jack van Lint, Discrete Math. 106/107 (1992), 303-309.
%H David A. Klarner and R. Rado, <a href="http://dx.doi.org/10.2140/pjm.1974.53.445">Arithmetic properties of certain recursively defined sets</a>, Pacific J. Math. 53 (1974), 445-463.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Klarner-RadoSequence.html">Klarner-Rado Sequence</a>.
%H <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%p ina:= proc(n) evalb(n=1) end:
%p a:= proc(n) option remember; local k, t;
%p if n=1 then 1
%p else for k from a(n-1)+1 while not
%p (irem(k, 2, 't')=0 and ina(t) or
%p irem(k, 3, 't')=2 and ina(t) or
%p irem(k, 6, 't')=3 and ina(t) )
%p do od: ina(k):= true; k
%p fi
%p end:
%p seq(a(n), n=1..80); # _Alois P. Heinz_, Mar 16 2011
%t s={1};Do[a=s[[n]];s=Union[s,{2a,3a+2,6a+3}],{n,1000}];s (* _Zak Seidov_, Mar 15 2011 *)
%t nxt[n_]:=Flatten[{#,2#,3#+2,6#+3}&/@n]; Take[Union[Nest[nxt,{1},5]],100] (* _Harvey P. Dale_, Feb 06 2015 *)
%o (C++)
%o #include <stdio.h>
%o #include <iostream>
%o #include <set>
%o using namespace std ;
%o int main(int argc, char *argv[])
%o { 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 ; }
%o - _R. J. Mathar_, Aug 20 2006
%o (Haskell)
%o import Data.Set (Set, fromList, insert, deleteFindMin)
%o a005658 n = a005658_list !! (n-1)
%o a005658_list = klarner $ fromList [1,2] where
%o klarner :: Set Integer -> [Integer]
%o klarner s = m : (klarner $
%o insert (2*m) $ insert (3*m+2) $ insert (6*m+3) s')
%o where (m,s') = deleteFindMin s
%o -- _Reinhard Zumkeller_, Mar 14 2011
%o (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
%Y Cf. A002977, A185661.
%K nonn,easy,nice
%O 1,2
%A _N. J. A. Sloane_
%E More terms from Larry Reeves (larryr(AT)acm.org), Oct 16 2000