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.

2, 4, 35, 8, 21, 70, 2055, 16, 8201, 42, 1035, 140, 141, 4110, 4111, 32, 529, 16402, 16403, 84, 85, 2070, 2071, 280, 65561, 282, 1180591620717411303451, 8220, 8221, 8222, 147573952589676412959, 64, 262177, 1058, 1059, 32804, 32805, 32806, 8388647, 168

Offset: 1

Author

Rémy Sigrist, Sep 29 2021

Keywords

• nonn

Comments

When a(n) > 0, the binary expansion of A125711(n) is a prefix of that of A125711(a(n)).

Examples

			The first terms, alongside the binary representations of A125711(n) and of A125711(a(n)), are:
  n  a(n)  h(n)               h(a(n))
  -  ----  -----------------  --------------------------------------
  1     2                  1                                      11
  2     4                 11                                     111
  3    35           10101111                          10101111101111
  4     8                111                                    1111
  5    21             101111                                10111111
  6    70          110101111                         110101111101111
  7  2055  10101011011101111  10101011011101111110111010101111101111
  8    16               1111                                   11111
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to 3x+1 (or Collatz) problem

Crossrefs

Cf. A006370, A006666, A070165, A125711.

Programs

Mathematica

A348094[n_] := n+2^(Length[NestWhileList[If[OddQ[#], 3#+1, #]/2 &, n, #>1 &]]-1);
Array[A348094, 50] (* Paolo Xausa, Apr 05 2024 *)

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 }

Formula

a(2^k) = 2^(k+1) for any k >= 0.

a(n) = n + 2^A006666(n) when A006666(n) >= 0.