A359226 a(n) is the least k >= 0 such that A006370^k(A070167(n)) = n (where A006370^k denotes the k-th iterate of A006370).

0, 0, 0, 5, 2, 0, 0, 4, 0, 1, 2, 0, 7, 2, 0, 3, 4, 0, 0, 9, 0, 1, 2, 0, 0, 6, 0, 1, 2, 0, 5, 2, 0, 3, 4, 0, 0, 2, 0, 8, 2, 0, 0, 4, 0, 1, 7, 0, 5, 2, 0, 5, 6, 0, 0, 2, 0, 1, 2, 0, 92, 4, 0, 1, 2, 0, 7, 2, 0, 3, 9, 0, 0, 7, 0, 1, 2, 0, 0, 8, 0, 1, 2, 0, 5, 2, 0

Offset: 1

Author

Rémy Sigrist, Dec 22 2022

Keywords

• nonn

Comments

If we travel the Collatz tree backwards, we will observe no further branching, whenever we have reached a number divisible by three. This is the reason why a(n) will be zero in all cases where n is divisible by three, because in this case no smaller number can exist further upward in the Collatz tree, as the actual branch will progress in numbers of the form 3*m*2^k. - Thomas Scheuerle, Dec 22 2022

Examples

			The Collatz sequence starting from 1 is: 1.
So a(1) = 0.
The Collatz sequence starting from 2 is: 2, 1.
So a(2) = 0.
The Collatz sequence starting from 3 is: 3, 10, 5, 16, 8, 4, 2, 1.
So a(3) = 0, a(10) = 1, a(5) = 2, a(16) = 3, a(8) = 4, a(4) = 5.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, PARI program
• Index entries for sequences related to 3x+1 (or Collatz) problem

Crossrefs

Cf. A006370, A070167, A177729.

Programs

Mathematica

nn = 87; c[] = -1; c[0] = 0; Do[(MapIndexed[If[c[#1] == -1, Set[c[#1], First[#2] - 1]] &, #]; -1 + Length[#]) &@ NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, c[#] == -1 && # > 1 &], {n, 0, nn}]; Array[c, nn] (* _Michael De Vlieger, Dec 23 2022 *)

PARI

See Links section.

Formula

a(n) = 0 iff n belongs to A177729.

If a(n) > 0 and n is odd, then a(n*3+1) - a(n) = 1. If a(n) > 0 and n is even, then a(n*3+1) - a(n*6+2) = 1. - Thomas Scheuerle, Dec 22 2022