A258772 Number of fixed points in the Collatz (3x+1) trajectory of n.

1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0

Offset

1,12

Comments

This sequence uses the definition in A006370: if n is odd, n -> 3n+1 and if n is even, n -> n/2.

The number 3 appears first at a(187561). Do all nonnegative numbers appear? See A258821.

Links

Paolo Xausa, Table of n, a(n) for n = 1..10000

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

Example

For n = 5, the trajectory is T(5) = [5, 16, 8, 4, 2, 1]. Since the fourth term in this sequence is 4, this is a fixed point. Since there is only one fixed point, a(5) = 1.
For n = 6, the trajectory is T(6) = [6, 3, 10, 5, 16, 8, 4, 2, 1]. Here, the k-th term in this trajectory does not equal k for any possible k. So a(6) = 0.

Mathematica

A258772[n_]:=Count[MapIndexed[{#1}==#2&, NestWhileList[If[OddQ[#], 3#+1, #/2]&, n, #>1&]], True]; Array[A258772, 100] (* Paolo Xausa, Nov 06 2023 *)

Prog

(PARI) Tvect(n)=v=[n]; while(n!=1, if(n%2, k=(3*n+1); v=concat(v, k); n=k); if(!(n%2), k=n/2; v=concat(v, k); n=k)); v
for(n=1, 200, d=Tvect(n); c=0; for(i=1, #d, if(d[i]==i, c++)); print1(c, ", "))

Crossrefs

Cf. A006370, A070165.

Sequence in context: A056620 A316869 A351961 * A178401 A357911 A280665

Adjacent sequences: A258769 A258770 A258771 * A258773 A258774 A258775

Keyword

nonn

Author

Derek Orr, Jun 09 2015

Status

approved