

A222118


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


4



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 nth row of A070165.  Andrey Zabolotskiy, Feb 27 2017


LINKS

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


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


CROSSREFS

Cf. A006577, A061641, A070165, A134191, A177729, A221956, A222297.
Sequence in context: A324398 A171916 A325740 * A267333 A186978 A228532
Adjacent sequences: A222115 A222116 A222117 * A222119 A222120 A222121


KEYWORD

nonn


AUTHOR

Jayanta Basu, Feb 23 2013


STATUS

approved



