A219696 Numbers k such that the trajectory of 3k + 1 under the '3x + 1' map reaches k.

1, 2, 4, 8, 10, 14, 16, 20, 22, 26, 40, 44, 52, 106, 184, 206, 244, 274, 322, 526, 650, 668, 790, 866, 976, 1154, 1300, 1438, 1732, 1780, 1822, 2308, 2734, 3238, 7288

Offset

1,2

Comments

This sequence seems complete; there are no other terms <= 10^9. - T. D. Noe, Dec 03 2012

If the 3x+1 step is replaced with (3x+1)/2, the sequence becomes {1, 2, 4, 8, 10, 14, 20, 22, 26, 40, 44, 206, 244, 650, 668, 866, 1154, 1822, 2308, ...}. - Robert G. Wilson v, Jan 13 2015

From Andrew Slattery, Aug 03 2023: (Start)

For most terms k, the trajectory of 3k + 1 reaches 310 or the trajectory of 310 reaches k.

For the rest of the terms k, the trajectory of 3k + 1 reaches 22 or the trajectory of 22 reaches k.

With the exception of k = 1, k is reached after S steps,

where S = c*8 + d*13 + e*44 + f*75, with c, d, e and f in {0, 1, 2}; in particular, S is in {8, 13, 8+8, 8+13, 13+13, 44, 75, 44+44, 75+13+13, 75+44, 75+75}. (End)

Links

Table of n, a(n) for n=1..35.

Eric Weisstein's World of Mathematics, Collatz Problem

Wikipedia, Collatz conjecture

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

Example

For k = 4, the Collatz trajectory of 3k + 1 is (13, 40, 20, 10, 5, 16, 8, 4, 2, 1), which includes 4; thus, 4 is in the sequence.
For k = 5, the Collatz trajectory of 3k + 1 is (16, 8, 4, 2, 1), which does not include 5; thus, 5 is not in the sequence.

Mathematica

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; Select[Range[10000], MemberQ[Collatz[3 # + 1], #] &] (* T. D. Noe, Dec 03 2012 *)

Prog

(Haskell)
a219696 n = a219696_list !! (n-1)
a219696_list = filter (\x -> collatz'' x == x) [1..] where
   collatz'' x = until (`elem` [1, x]) a006370 (3 * x + 1)
-- Reinhard Zumkeller, Aug 11 2014
(Python)
def ok(n):
    if n==1: return [1]
    N=3*n + 1
    l=[N, ]
    while True:
        if N%2==1: N = 3*N + 1
        else: N/=2
        l+=[N, ]
        if N<2: break
    if n in l: return 1
    return 0 # Indranil Ghosh, Apr 22 2017
(PARI) a006370(n) = if(n%2==0, n/2, 3*n+1)
is(n) = my(x=3*n+1); while(1, x=a006370(x); if(x==n, return(1), if(x==1, return(0)))) \\ Felix Fröhlich, Jun 10 2021

Crossrefs

Cf. A014682, A070991, A006370, A070165.

Sequence in context: A328588 A356435 A287844 * A087505 A086801 A154115

Adjacent sequences: A219693 A219694 A219695 * A219697 A219698 A219699

Keyword

nonn,nice,more

Author

Robert C. Lyons, Nov 25 2012

Extensions

Initial 1 from Clark R. Lyons, Dec 02 2012

Status

approved