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%I #35 Aug 31 2017 14:25:05
%S 1,3,4,7,9,10,13,15,19,21,22,25,27,28,31,39,40,43,45,46,51,55,57,58,
%T 61,63,64,67,76,79,81,82,85,87,91,93,94,103,111,115,117,118,121,123,
%U 127,129,130,133,135,136,139,151,153,154,159,163,165,166,169,171,172,175,183,184,187,189,190,193,202
%N Smallest set containing 1 and closed under the operations x->2x+1, x->3x+1, x->6x+1.
%C This sequence has density zero.
%C To illustrate the density: there are 2011 terms up to 10^4, 14878 terms up to 10^5, 117671 terms up to 10^6, 913314 terms up to 10^7, 7176461 terms up to 10^8, 56591334 terms up to 10^9, and 445290307 terms up to 10^10. - _Charles R Greathouse IV_, Jul 09 2017
%C There are 3560822110 terms up to 10^11, 27907016447 terms up to 10^12, 223533750957 terms up to 10^13, 1772572144707 terms up to 10^14, ..., roughly 7.952916868743154^m/log(10) terms up to 10^m. - _Yi Yang_, Aug 29 2017
%D J. C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, Amer. Math. Soc., 2010. See pp. 6, 280.
%H G. C. Greubel, <a href="/A185661/b185661.txt">Table of n, a(n) for n = 1..5000</a>
%H <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%t terms = 69; Clear[f]; f[n_] := f[n] = With[{lst = NestList[{2 # + 1, 3 # + 1, 6 # + 1}&, 1, n] // Flatten // Union}, If[Length[lst] <= terms, lst, Take[lst, terms]]]; f[1]; f[n = 2]; While[f[n] != f[n-1], Print["n = ", n]; n++]; A185661 = f[n] (* _Jean-François Alcover_, May 17 2017 *)
%o (PARI) list(lim)=my(v=List([1]),i,t); while(i++<=#v, t=2*v[i]+1; if(t>lim, next); listput(v,t); t+=v[i]; if(t>lim, next); listput(v,t); t+=t-1; if(t>lim, next); listput(v,t)); Set(v) \\ _Charles R Greathouse IV_, Jul 09 2017
%o (PARI) list(lim)=my(v=List([1]),m=Map(),t,i); while(i++<=#v, t=2*v[i]+1; if(t>lim, next); if(!mapisdefined(m,t), mapput(m,t,0); listput(v,t)); t+=v[i]; if(t>lim, next); if(!mapisdefined(m,t), mapput(m,t,0); listput(v,t)); t+=t-1; if(t<=lim && !mapisdefined(m,t), mapput(m,t,0); listput(v,t))); m=0; Set(v) \\ _Charles R Greathouse IV_, Jul 09 2017
%Y Cf. A002977.
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_, Feb 08 2011
%E Name clarified by _Charles R Greathouse IV_, Jul 09 2017
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