A304715 For any n > 0, if A006666(n) >= 0, then a(n) = Sum_{i = 0..A006666(n)-1} 2^i * [T^i(n) == 0 (mod 2)] (where [] is an Iverson bracket and T^i denotes the i-th iterate of the Collatz function A014682); otherwise a(n) = -1.

0, 1, 28, 3, 14, 57, 1896, 7, 7586, 29, 948, 115, 118, 3793, 3824, 15, 474, 15173, 15180, 59, 62, 1897, 1912, 231, 60722, 237, 1102691417057682138372, 7587, 7590, 7649, 137836427132210267296, 31, 242890, 949, 956, 30347, 30350, 30361, 7772616, 119

Offset

1,3

Comments

In other words, when a(n) >= 0, the binary representation of a(n) encodes the tripling and halvings steps of the Collatz compressed trajectory of n up to the first occurrence of the number 1 (where zeros and ones respectively denote tripling and halving steps).

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Index entries for sequences related to 3x+1 (or Collatz) problem

Formula

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

a(2*n) = 2*a(n) + 1.

A029837(a(n)+1) = A006666(n).

A000120(a(n)) = A220071(n).

a(A248573(n)) < a(A248573(n+1)) for any n >= 0. - Rémy Sigrist, Nov 09 2018

Example

The first terms, alongside the binary representation of a(n) and the Collatz compressed trajectory of a(n) up to the first 1 in reverse order, are:
  n    a(n)       bin(a(n))  rev(traj(n))
  --   ----       ---------  ------------
   1      0               0  (1)
   2      1               1  (1, 2)
   3     28           11100  (1, 2, 4, 8, 5, 3)
   4      3              11  (1, 2, 4)
   5     14            1110  (1, 2, 4, 8, 5)
   6     57          111001  (1, 2, 4, 8, 5, 3, 6)
   7   1896     11101101000  (1, 2, 4, 8, 5, 10, 20, 13, 26, 17, 11, 7)
   8      7             111  (1, 2, 4, 8)
   9   7586   1110110100010  (1, 2, 4, 8, 5, 10, 20, 13, 26, 17, 11, 7, 14, 9)
  10     29           11101  (1, 2, 4, 8, 5, 10)
  11    948      1110110100  (1, 2, 4, 8, 5, 10, 20, 13, 26, 17, 11)
  12    115         1110011  (1, 2, 4, 8, 5, 3, 6, 12)
  13    118         1110110  (1, 2, 4, 8, 5, 10, 20, 13)
  14   3793    111011010001  (1, 2, 4, 8, 5, 10, 20, 13, 26, 17, 11, 7, 14)
  15   3824    111011110000  (1, 2, 4, 8, 5, 10, 20, 40, 80, 53, 35, 23, 15)
  16     15            1111  (1, 2, 4, 8, 16)
  17    474       111011010  (1, 2, 4, 8, 5, 10, 20, 13, 26, 17)
  18  15173  11101101000101  (1, 2, 4, 8, 5, 10, 20, 13, 26, 17, 11, 7, 14, 9, 18)

Prog

(PARI) a(n) = my (v=0); for (k=0, oo, if (n==1, return (v), n%2, n = (3*n+1)/2, n = n/2; v += 2^k))

Crossrefs

Cf. A000120, A006666, A014682, A029837, A220071, A248573.

Sequence in context: A040770 A040772 A040773 * A040767 A040768 A040769

Adjacent sequences: A304712 A304713 A304714 * A304716 A304717 A304718

Keyword

nonn,base,look

Author

Rémy Sigrist, May 17 2018

Status

approved