A222118 Number of terms in Collatz (3x+1) trajectory of n that did not appear in previous trajectories.

1, 1, 6, 0, 0, 1, 10, 0, 3, 0, 0, 1, 0, 0, 9, 0, 0, 1, 5, 0, 3, 0, 0, 1, 3, 0, 95, 0, 0, 1, 0, 0, 3, 0, 0, 1, 3, 0, 12, 0, 0, 1, 8, 0, 3, 0, 0, 1, 0, 0, 5, 0, 0, 1, 7, 0, 3, 0, 0, 1, 0, 0, 13, 0, 0, 1, 0, 0, 3, 0, 0, 1, 3, 0, 8, 0, 0, 1, 9, 0, 1, 0, 0, 1, 0, 0, 7

Offset: 1

Author

Jayanta Basu, Feb 23 2013

Keywords

• nonn

Comments

For n > 2, n such that a(n) = 0 are termed impure (A134191), while n such that a(n) > 0 are termed pure (A061641). - T. D. Noe, Feb 23 2013

From Robert G. Wilson v, Feb 25 2017: (Start)

For a(n) to be equal to 0, n != 0 (mod 3),

For a(n) to be an even positive number, n = {3, 7} (mod 12),

For a(n) to be equal to 1, n = {0, 1, 2, 3, 6, 7, 9} (mod 12),

For a(n) to be equal to 3, n = {1, 3, 9} (mod 12),

For a(n) to be an odd number > 3, n = {3, 7} (mod 12).

[Note that the above conditions are necessary but not sufficient. - Editors, Dec 15 2017]

(End)

a(n) gives the number of new terms in the n-th row of A070165 (see A263716). - Andrey Zabolotskiy, Feb 27 2017

Examples

			a(7) = 10, since trajectory of 7 includes 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, which did not appear in earlier trajectories.
		

Links

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

Crossrefs

Cf. A006577, A061641, A070165, A134191, A177729, A221956, A222297, A263716.

Programs

Mathematica

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; found = {}; Table[c = Collatz[n]; r = Complement[c, found]; found = Union[found, c]; Length[r], {n, 100}] (* T. D. Noe, Feb 23 2013 *)

Python

s = set([1])
print(1)
for n in range(2, 100):
    m, r = n, 0
    while m not in s:
        s.add(m)
        m = (m//2 if m%2==0 else 3*m+1)
        r += 1
    print(r)
# Andrey Zabolotskiy, Feb 21 2017

Formula

a(n) = A006577(n) - A221956(n) + 1. - Michel Lagneau, Feb 23 2013