A320538 Assuming the truth of the Collatz conjecture, a(n) is the number of divisors of n appearing in the Collatz trajectory of n.

1, 2, 2, 3, 2, 4, 2, 4, 2, 4, 2, 6, 2, 4, 3, 5, 2, 4, 2, 6, 2, 4, 2, 8, 3, 4, 2, 6, 2, 6, 2, 6, 3, 4, 3, 6, 2, 4, 3, 8, 2, 4, 2, 6, 3, 4, 2, 10, 3, 6, 3, 6, 2, 4, 3, 8, 2, 4, 2, 9, 2, 4, 2, 7, 4, 6, 2, 6, 2, 6, 2, 8, 2, 4, 2, 6, 3, 6, 2, 10, 2, 4, 2, 6, 2, 4, 2

Offset: 1

Author

Michel Lagneau, Oct 15 2018

Keywords

• nonn

Comments

a(p) = 2 for p prime.

a((2^2k - 1)/3) = 2, k = 1, 2, ...

We observe that a(n) differs from A093640(n) for n = 25, 27, 33, 35, 45, 49, 50, 54, 55, 57, 63, 65, 66, 70, 75, 77, 85, ...

7 occurs only eighteen times among the first 65537 terms. - Antti Karttunen, May 18 2019

Examples

			a(6) = 4 because the Collatz trajectory 6 -> 3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1 contains 4 divisors of 6: 1, 2, 3 and 6.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences related to 3x+1 (or Collatz) problem

Crossrefs

Cf. A006370, A027750, A070165, A093640, A207674, A207675.

Programs

Mathematica

lst={}; coll[n_]:=NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&]; Do[AppendTo[lst,Length[Intersection[Divisors[n],coll[n]]]],{n,1,100}]; lst

PARI

f(n) = if(n%2, 3*n+1, n/2);
a(n) = {my(kn = n, nb = 1); while (n != 1, n = f(n); if ((kn % n) == 0, nb++);); nb;}