A258822 Number of times that k iterations of n under the '3x+1' map yield k for some k.

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

Offset: 1

Author

Derek Orr, Jun 11 2015

Keywords

• nonn

Comments

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

The number 3 appears first at a(63105). Do all nonnegative numbers appear? See A258824.

Examples

			For n = 6, the '3x+1' map is as follows: 6 -> 3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1. Here the number of iterations is 8. However, after the k-th iteration, the result does not equal k. Thus a(6) = 0.
For n = 7, the '3x+1' map is as follows: 7 -> 22 -> 11 -> 34 -> 17 -> 52 -> 26 -> 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1. Only after 10 iterations do we arrive at 10. Since this is the only time this happens, a(7) = 1.
		

Links

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

Crossrefs

Cf. A006370, A258772, A070165.

Programs

Mathematica

A258822[n_]:=Count[MapIndexed[{#1}==#2-1&,NestWhileList[If[OddQ[#],3#+1,#/2]&,n,#>1&]],True];Array[A258822,100] (* Paolo Xausa, Nov 06 2023 *)

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-1, c++)); print1(c, ", "))