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, 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.
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
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Oct 16 2000
STATUS
approved