A303698 Numbers k whose trajectory in the Collatz (or '3x+1') problem includes another multiple of k.

31, 62, 83, 166, 293, 347, 586, 671, 694, 1342, 2684, 19151, 38302, 2025797, 4051594

Offset: 1

Author

Jon E. Schoenfield, Apr 28 2018

Keywords

• nonn
• more

Comments

If a term k is even, then k/2 is also in the sequence.

For each odd term k among the first 15 terms, 2*k is also a term. Does the sequence include any odd terms k for which 2*k is not also in the sequence?

If it exists, a(16) > 1.5*10^11. - Giovanni Resta, May 04 2018

Examples

			The Collatz trajectory of 31 begins 31 -> 94 -> 47 -> 142 -> 71 -> 214 -> 107 -> 322 -> 161 -> 484 -> 242 -> 121 -> 364 -> 182 -> 91 -> 274 -> 137 -> 412 -> 206 -> 103 -> 310 -> 155 -> 466 -> ... which contains 310 and 155, both of which are multiples of 31, so 31 is in the sequence.
Other than its initial value, the trajectory of 62 is the same as that of 31, so it also includes 310, which is a multiple of 62, so 62 is in the sequence.
The trajectory of 671 includes 29524 = 671 * 11 * 2^2, so the sequence includes 671, 671*2 = 1342, and 671*4 = 2684.
		

Links

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

Crossrefs

Cf. A005184, A006370, A070165.

Programs

Magma

a:=[]; for k in [2..40000] do t:=k; while t gt 1 do if IsEven(t) then t:=t div 2; else t:=3*t+1; end if; if IsDivisibleBy(t,k) then a[#a+1]:=k; break; end if; end while; end for; a; // Jon E. Schoenfield, Apr 30 2018

Maple

T:= proc(n) option remember; `if`(n=1, 1,
      [n, T(`if`(n::even, n/2, 3*n+1))][])
    end:
q:= n-> ormap(x-> x>n and irem(x, n)=0, [T(n)]):
select(q, [$1..40000])[];  # Alois P. Heinz, Aug 04 2025

Mathematica

traj[1]:={1};
traj[n_]:=traj[n]=If[EvenQ[n]&&n>0,{n}~Join~traj[n/2],{n}~Join~traj[3*n+1]];
fQ[n_]:=Select[traj[n],IntegerQ[#/n]&&#/n>1&,1]!={};
Select[Range[20000],fQ[#]&] (* Ivan N. Ianakiev, Apr 30 2018 *)