A213181 Number of chains of even numbers of length 2 or more in the Collatz (3x+1) trajectory of n.

0, 0, 1, 1, 1, 1, 3, 1, 4, 1, 3, 2, 2, 3, 2, 1, 3, 4, 4, 2, 1, 3, 2, 2, 5, 2, 17, 4, 4, 2, 16, 1, 6, 3, 2, 5, 4, 4, 6, 2, 17, 1, 6, 4, 4, 2, 16, 2, 5, 5, 5, 3, 2, 17, 17, 4, 7, 4, 6, 3, 3, 16, 15, 1, 6, 6, 5, 4, 3, 2, 16, 5, 18, 4, 2, 5, 5, 6, 6, 2, 4, 17, 17

Offset: 1

Author

Jayanta Basu, Feb 28 2013

Keywords

• nonn

Comments

A pair of even numbers that appear side by side in Collatz trajectory of n is considered a chain of length 2 and likewise for chains of greater length.

Examples

			For n=3, Collatz trajectory of 3 is 3,10,5,16,8,4,2,1, hence the only chain is 16,8,4,2 and so a(3)=1.
For n=12: 12,6,3,10,5,16,8,4,2,1 and as such there are two chains 12,6 and 16,8,4,2 so a(12)=2.
		

Links

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

Crossrefs

Cf. A014682, A070165, A133872, A139391, A363270.

Programs

Mathematica

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; Table[c = Collatz[n]; cnt = 0; evenCnt = 0; Do[If[OddQ[i], evenCnt = 0, evenCnt++; If[evenCnt == 2, cnt++]], {i, c}]; cnt, {n, 100}] (* T. D. Noe, Feb 28 2013 *)

Formula

a(n) = a(A363270(A014682(n))) + 1 for n >= 3. - Alan Michael Gómez Calderón, Apr 09 2025

a(n) = a(A139391(n)) + A133872(n) for n >= 2. - Alan Michael Gómez Calderón, Apr 23 2025