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A323097
Numbers m such that all elements of the Collatz trajectory occur in the divisors of m.
0
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
1,2
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.
EXAMPLE
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
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 *)
KEYWORD
nonn
AUTHOR
Michel Lagneau, Aug 30 2019
STATUS
approved