A221956 -id:A221956 - cp's OEIS frontend

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

Jayanta Basu, Feb 23 2013

nonn

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

			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.

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

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 *)

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

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

cp's OEIS Frontend

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

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