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%I #13 Oct 05 2019 10:48:00
%S 1,2,4,8,16,32,64,80,128,160,256,320,512,640,1024,1280,1344,2048,2560,
%T 2688,4096,5120,5376,8192,10240,10752,16384,20480,21504,21760,32768,
%U 40960,43008,43520,65536,81920,86016,87040,131072,163840,172032,174080,262144,327680
%N Numbers m such that all elements of the Collatz trajectory occur in the divisors of m.
%C See A207674 (numbers such that all divisors occur in their Collatz trajectories).
%C The powers of 2 are in the sequence.
%C 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.
%C The numbers 5*2^k (A020714) for k > 3 are in the sequence.
%C The numbers 21*2^k (A175805) for k > 5 are in the sequence.
%C The numbers 85*2^k for k > 7 are in the sequence.
%C 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.
%e 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
%p with(numtheory):nn:=250000:
%p for n from 1 to nn do:
%p m:=n:it:=0:lst:={n}:
%p for i from 1 to nn while(m<>1) do:
%p if irem(m, 2)=0
%p then
%p m:=m/2:
%p else
%p m:=3*m+1:
%p fi:
%p it:=it+1:lst:=lst union {m}:
%p od:
%p x:=divisors(n):n0:=nops(x):lst1:={op(x), x[n0]}:
%p lst2:=lst intersect lst1:n1:=nops(lst2):
%p if lst2=lst
%p then
%p printf(`%d, `,n):
%p else fi:
%p od:
%t aQ[n_] := n == LCM @@ NestWhileList[If[OddQ[#], 3 # + 1, #/2] &, n, # > 1 &]; Select[Range[330000], aQ] (* _Amiram Eldar_, Aug 31 2019 *)
%Y Cf. A000079, A006370, A006577, A020714, A027750, A070165, A175805, A207674, A272985.
%K nonn
%O 1,2
%A _Michel Lagneau_, Aug 30 2019
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