login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A219696
Numbers k such that the trajectory of 3k + 1 under the '3x + 1' map reaches k.
2
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)
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
KEYWORD
nonn,nice,more
AUTHOR
Robert C. Lyons, Nov 25 2012
EXTENSIONS
Initial 1 from Clark R. Lyons, Dec 02 2012
STATUS
approved