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

1,2

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

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences related to 3x+1 (or Collatz) problem

Example

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.

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

Prog

(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; }

Crossrefs

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

Sequence in context: A025479 A079243 A093640 * A343650 A327391 A365207

Adjacent sequences: A320535 A320536 A320537 * A320539 A320540 A320541

Keyword

nonn

Author

Michel Lagneau, Oct 15 2018

Status

approved