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A116623 sorted, without duplicates.
6

%I #12 Sep 15 2019 11:15:53

%S 1,5,7,11,19,23,29,31,35,37,47,49,53,65,67,73,79,85,89,97,101,103,119,

%T 121,125,131,133,143,149,151,157,161,169,175,179,185,197,205,211,215,

%U 221,223,227,233,239,251,259,269,271,275,277,283,287,289,313,319,323

%N A116623 sorted, without duplicates.

%C Related to the parity vectors of Terras and Collatz trajectories.

%C From _Bob Selcoe_, Sep 14 2019: (Start)

%C Let R_s be the reduced Collatz sequence starting with s and let R_s(i), i >= 0 be the i-th term in R_s. Then any term in R_s can be described as (3*s^i + k)/2^j, where j is the total number of halving steps from R_s(0) to R_s(i) i >= 1, and k is some term in A116641. k=1 when i=1; when i > 1, k is determined by the specific order of halving steps in R_s.

%C Ignoring duplicates, terms in A116641 > 1 can be generated by a series of subsequences:

%C 1. Start with subsequence a(m) = 3+2^m, m >= 1; i.e., a(m) = {5,7,11,19,35,67,...}.

%C 2. For fixed m, generate new subsequences b(n) = 3*a(m) + 2^(m+n), n >= 1; so:

%C m=1, a(1)=5, b(n) = 3*5 + {4,8,16,32,...} = {19,23,31,47,...};

%C m=2, a(2)=7, b(n) = 3*7 + {8,16,32,64,...} = {29,37,53,85,...};

%C m=3, a(3)=11, b(n) = 3*11 + {16,32,64,128,...} = {49,65,97,161,...}; etc.

%C 3. Let 2^y be the summand used to find terms (t) in any previously-generated subsequence. (For instance, in m=2, b(3)=53: y=5 because t=53 = 3*7 + 32.) Continue generating new subsequences p(q) = 3*t + 2^(y+z) {z=1..inf} for all t. So in this example, from t=53 we get p(q) = 3*53 + {64,128,256,512,...} = {223,287,415,671,...}; from t=671 we get p(q) = 3*671 + {1024,2048,4096,...} = {3037,4061,6109,...), etc.

%C (End)

%H <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>

%Y Cf. A116642 gives the same sequence in binary.

%K nonn

%O 0,2

%A _Antti Karttunen_, Feb 20 2006. Proposed by Pierre Lamothe (plamothe(AT)aei.ca), May 21 2004.