A222118 - OEIS

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

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OFFSET

1,3

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

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

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

FORMULA

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

EXAMPLE

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.

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 *)

PROG

(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