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

Author

Rémy Sigrist, May 17 2018

Keywords

• nonn
• base
• look

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).

Examples

			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)
		

Links

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

Crossrefs

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

Programs

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))

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