A323097 Numbers m such that all elements of the Collatz trajectory occur in the divisors of m.

1, 2, 4, 8, 16, 32, 64, 80, 128, 160, 256, 320, 512, 640, 1024, 1280, 1344, 2048, 2560, 2688, 4096, 5120, 5376, 8192, 10240, 10752, 16384, 20480, 21504, 21760, 32768, 40960, 43008, 43520, 65536, 81920, 86016, 87040, 131072, 163840, 172032, 174080, 262144, 327680

Offset: 1

Author

Michel Lagneau, Aug 30 2019

Keywords

• nonn

Comments

See A207674 (numbers such that all divisors occur in their Collatz trajectories).

The powers of 2 are in the sequence.

The number 80 is probably the unique non-power of 2 of the sequence such that the elements of the Collatz trajectory are exactly the same as the divisors.

The numbers 5*2^k (A020714) for k > 3 are in the sequence.

The numbers 21*2^k (A175805) for k > 5 are in the sequence.

The numbers 85*2^k for k > 7 are in the sequence.

In the general case, the numbers of the form ((4^i - 1)/3)*2^j for i = 1, 2,... and j = 2i, 2i+1, 2i+2, ... are in the sequence.

Examples

			1344 is in the sequence because the set of the divisors {1, 2, 3, 4, 6, 7, 8, 12, 14, 16, 21, 24, 28, 32, 42, 48, 56, 64, 84, 96, 112, 168, 192, 224, 336, 448, 672, 1344} contains the set of the elements of the Collatz trajectory 1344 -> 672 -> 336 -> 168 -> 84 -> 42 -> 21 -> 64 -> 32 -> 16 -> 8 -> 4 -> 2 -> 1
		

Crossrefs

Cf. A000079, A006370, A006577, A020714, A027750, A070165, A175805, A207674, A272985.

Programs

Maple

with(numtheory):nn:=250000:
  for n from 1 to nn do:
    m:=n:it:=0:lst:={n}:
      for i from 1 to nn while(m<>1) do:
        if irem(m, 2)=0
         then
         m:=m/2:
         else
         m:=3*m+1:
        fi:
       it:=it+1:lst:=lst union {m}:
      od:
       x:=divisors(n):n0:=nops(x):lst1:={op(x), x[n0]}:
       lst2:=lst intersect lst1:n1:=nops(lst2):
       if lst2=lst
       then
       printf(`%d, `,n):
       else fi:
     od:

Mathematica

aQ[n_] := n == LCM @@ NestWhileList[If[OddQ[#], 3 # + 1, #/2] &, n, # > 1 &]; Select[Range[330000], aQ] (* Amiram Eldar, Aug 31 2019 *)