cp's OEIS Frontend

A348094 If the Collatz trajectory of n reaches 1, say after k steps, and there is an integer m > n such that T^i(m) and T^i(n) have the same parity for i = 0..k (where T^i denotes the i-th iterate of the Collatz map A006370), then a(n) is the least such m, otherwise a(n) is -1.

%I A348094 #14 Apr 05 2024 10:07:14
%S A348094 2,4,35,8,21,70,2055,16,8201,42,1035,140,141,4110,4111,32,529,16402,
%T A348094 16403,84,85,2070,2071,280,65561,282,1180591620717411303451,8220,8221,
%U A348094 8222,147573952589676412959,64,262177,1058,1059,32804,32805,32806,8388647,168
%N A348094 If the Collatz trajectory of n reaches 1, say after k steps, and there is an integer m > n such that T^i(m) and T^i(n) have the same parity for i = 0..k (where T^i denotes the i-th iterate of the Collatz map A006370), then a(n) is the least such m, otherwise a(n) is -1.
%C A348094 When a(n) > 0, the binary expansion of A125711(n) is a prefix of that of A125711(a(n)).
%H A348094 Paolo Xausa, <a href="/A348094/b348094.txt">Table of n, a(n) for n = 1..10000</a>
%H A348094 <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%F A348094 a(2^k) = 2^(k+1) for any k >= 0.
%F A348094 a(n) = n + 2^A006666(n) when A006666(n) >= 0.
%e A348094 The first terms, alongside the binary representations of A125711(n) and of A125711(a(n)), are:
%e A348094   n  a(n)  h(n)               h(a(n))
%e A348094   -  ----  -----------------  --------------------------------------
%e A348094   1     2                  1                                      11
%e A348094   2     4                 11                                     111
%e A348094   3    35           10101111                          10101111101111
%e A348094   4     8                111                                    1111
%e A348094   5    21             101111                                10111111
%e A348094   6    70          110101111                         110101111101111
%e A348094   7  2055  10101011011101111  10101011011101111110111010101111101111
%e A348094   8    16               1111                                   11111
%t A348094 A348094[n_] := n+2^(Length[NestWhileList[If[OddQ[#], 3#+1, #]/2 &, n, #>1 &]]-1);
%t A348094 Array[A348094, 50] (* _Paolo Xausa_, Apr 05 2024 *)
%o A348094 (PARI) a(n) = { my (h=0, r=n); while (r>1, if (r%2, r=3*r+1, r=r/2; h++)); n+2^h }
%Y A348094 Cf. A006370, A006666, A070165, A125711.
%K A348094 nonn
%O A348094 1,1
%A A348094 _Rémy Sigrist_, Sep 29 2021