OFFSET
1,3
COMMENTS
The original name was: Number of iterations of the Collatz recursion required to reach a power of 2.
The statement that all paths must eventually reach a power of 2 is equivalent to the Collatz conjecture.
A006577(n) - a(n) gives the exponent for the first power of 2 reached in the Collatz trajectory of n. - Alonso del Arte, Mar 05 2012
Number of nonpowers of 2 in the 3x+1 sequence starting at n. - Omar E. Pol, Sep 05 2021
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
John Smith, Collatz sequence, PlanetMath.
Eric Weisstein's World of Mathematics, Collatz problem.
FORMULA
For x>0 an integer, define f_0(x)=x, and for r=1,2,..., f_r(x)=f_{r-1}(x)/2 if f_{r-1}(x) is even, else f_r(x)=3*f_{r-1}(x)+1. Then a(n) = min(k such that f_k(n) is equal to a power of 2).
a(n) = A006577(n) - A135282(n) (after Alonso del Arte's comment), if A006577(n) is not -1. - Omar E. Pol, Apr 10 2022
EXAMPLE
a(7) = 12 because the Collatz trajectory for 7 is 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, ... which reached 16 = 2^4 in 12 steps.
MAPLE
a:= proc(n) option remember; `if`(n=2^ilog2(n), 0,
1+a(`if`(n::odd, 3*n+1, n/2)))
end:
seq(a(n), n=1..100); # Alois P. Heinz, Sep 05 2021
MATHEMATICA
Collatz[n_?OddQ] := 3*n + 1; Collatz[n_?EvenQ] := n/2; Table[-1 + Length[NestWhileList[Collatz, n, Not[IntegerQ[Log[2, #]]] &]], {n, 50}] (* Alonso del Arte, Mar 04 2012 *)
PROG
(Haskell)
a208981 = length . takeWhile ((== 0) . a209229) . a070165_row
-- Reinhard Zumkeller, Jan 02 2013
(PARI) ispow2(n)=n>>=valuation(n, 2); n==1
a(n)=my(s); while(!ispow2(n), n=if(n%2, 3*n+1, n/2); s++); s \\ Charles R Greathouse IV, Jul 31 2016
CROSSREFS
KEYWORD
AUTHOR
L. Edson Jeffery, Mar 04 2012
EXTENSIONS
Name clarified by Omar E. Pol, Apr 10 2022
STATUS
approved