A220421 Number of halving and tripling steps to reach the largest value in the Collatz (3x+1) trajectory of n.

0, 0, 3, 0, 1, 4, 5, 0, 8, 2, 3, 5, 1, 6, 7, 0, 1, 9, 3, 0, 1, 4, 5, 0, 6, 2, 77, 7, 1, 8, 72, 0, 1, 2, 3, 10, 1, 4, 10, 0, 75, 2, 3, 5, 1, 6, 70, 0, 1, 7, 3, 0, 1, 78, 78, 0, 6, 2, 8, 9, 1, 73, 73, 0, 1, 2, 3, 0, 1, 4, 68, 0, 81, 2, 3, 5, 1, 11, 7, 0, 1, 76

Offset: 1

Author

Jayanta Basu, Feb 19 2013

Keywords

• nonn
• changed

Comments

a(n) = 0 if n is a power of 2, as a(1) = a(2) = a(4) = ... = 0; however a(20) = a(24) = ... = 0 also and as such the condition (n = 2^k, k>=0) is sufficient but not necessary for a(n) = 0.

Examples

			a(3) = 3 because the Collatz trajectory for 3 is [3, 10, 5, 16, 8, 4, 2, 1], reaching the largest term, 16, in three steps.
a(4) = 0 because the Collatz trajectory only goes down from 4.
a(20) = 0: 20 is the largest term in [20, 10, 5, 16, 8, 4, 2, 1].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A006577, A014682, A025586.

Programs

Mathematica

Collatz[n_] :=NestWhileList[If[EvenQ[#], #/2, 3*# + 1] &, n, # > 1 &]; Table[Position[Collatz[n], Max[Collatz[n]]][[1, 1]] - 1, {n, 82}] (* Jayanta Basu, Mar 24 2013 *)

Formula

a(n) = A087225(n) - 1.

Extensions

More terms from Alois P. Heinz, Feb 20 2013