cp's OEIS Frontend

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

%I A303698 #26 Aug 04 2025 18:52:47
%S A303698 31,62,83,166,293,347,586,671,694,1342,2684,19151,38302,2025797,
%T A303698 4051594
%N A303698 Numbers k whose trajectory in the Collatz (or '3x+1') problem includes another multiple of k.
%C A303698 If a term k is even, then k/2 is also in the sequence.
%C A303698 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?
%C A303698 If it exists, a(16) > 1.5*10^11. - _Giovanni Resta_, May 04 2018
%H A303698 <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%e A303698 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.
%e A303698 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.
%e A303698 The trajectory of 671 includes 29524 = 671 * 11 * 2^2, so the sequence includes 671, 671*2 = 1342, and 671*4 = 2684.
%p A303698 T:= proc(n) option remember; `if`(n=1, 1,
%p A303698       [n, T(`if`(n::even, n/2, 3*n+1))][])
%p A303698     end:
%p A303698 q:= n-> ormap(x-> x>n and irem(x, n)=0, [T(n)]):
%p A303698 select(q, [$1..40000])[];  # _Alois P. Heinz_, Aug 04 2025
%t A303698 traj[1]:={1};
%t A303698 traj[n_]:=traj[n]=If[EvenQ[n]&&n>0,{n}~Join~traj[n/2],{n}~Join~traj[3*n+1]];
%t A303698 fQ[n_]:=Select[traj[n],IntegerQ[#/n]&&#/n>1&,1]!={};
%t A303698 Select[Range[20000],fQ[#]&] (* _Ivan N. Ianakiev_, Apr 30 2018 *)
%o A303698 (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
%Y A303698 Cf. A005184, A006370, A070165.
%K A303698 nonn,more
%O A303698 1,1
%A A303698 _Jon E. Schoenfield_, Apr 28 2018