cp's OEIS Frontend

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

%I A323097 #13 Oct 05 2019 10:48:00
%S A323097 1,2,4,8,16,32,64,80,128,160,256,320,512,640,1024,1280,1344,2048,2560,
%T A323097 2688,4096,5120,5376,8192,10240,10752,16384,20480,21504,21760,32768,
%U A323097 40960,43008,43520,65536,81920,86016,87040,131072,163840,172032,174080,262144,327680
%N A323097 Numbers m such that all elements of the Collatz trajectory occur in the divisors of m.
%C A323097 See A207674 (numbers such that all divisors occur in their Collatz trajectories).
%C A323097 The powers of 2 are in the sequence.
%C A323097 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 A323097 The numbers 5*2^k (A020714) for k > 3 are in the sequence.
%C A323097 The numbers 21*2^k (A175805) for k > 5 are in the sequence.
%C A323097 The numbers 85*2^k for k > 7 are in the sequence.
%C A323097 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 A323097 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 A323097 with(numtheory):nn:=250000:
%p A323097   for n from 1 to nn do:
%p A323097     m:=n:it:=0:lst:={n}:
%p A323097       for i from 1 to nn while(m<>1) do:
%p A323097         if irem(m, 2)=0
%p A323097          then
%p A323097          m:=m/2:
%p A323097          else
%p A323097          m:=3*m+1:
%p A323097         fi:
%p A323097        it:=it+1:lst:=lst union {m}:
%p A323097       od:
%p A323097        x:=divisors(n):n0:=nops(x):lst1:={op(x), x[n0]}:
%p A323097        lst2:=lst intersect lst1:n1:=nops(lst2):
%p A323097        if lst2=lst
%p A323097        then
%p A323097        printf(`%d, `,n):
%p A323097        else fi:
%p A323097      od:
%t A323097 aQ[n_] := n == LCM @@ NestWhileList[If[OddQ[#], 3 # + 1, #/2] &, n, # > 1 &]; Select[Range[330000], aQ] (* _Amiram Eldar_, Aug 31 2019 *)
%Y A323097 Cf. A000079, A006370, A006577, A020714, A027750, A070165, A175805, A207674, A272985.
%K A323097 nonn
%O A323097 1,2
%A A323097 _Michel Lagneau_, Aug 30 2019