A207674 -id:A207674 - cp's OEIS frontend

A207675 Numbers such that not all divisors occur in their Collatz trajectories.

9, 15, 18, 21, 27, 30, 33, 35, 36, 39, 42, 45, 51, 54, 55, 57, 60, 63, 66, 69, 70, 72, 75, 77, 78, 81, 84, 85, 87, 90, 91, 93, 95, 96, 99, 102, 105, 108, 110, 111, 114, 115, 117, 119, 120, 121, 123, 125, 126, 129, 132, 133, 135, 138, 140, 141, 143, 144, 145

Offset: 1

Reinhard Zumkeller, Feb 20 2012

nonn

			3 is a divisor of 9, not occurring in A033479 - therefore 9 is a term.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Collatz Problem
Wikipedia, Collatz conjecture
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A027750, A070165, A006370, A207674 (complement).

import Data.List (intersect)
a207675 n = a207675_list !! (n-1)
a207675_list = filter
   (\x -> a027750_row x `intersect` a070165_row x /= a027750_row x) [1..]

coll[n_]:=NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&]; Select[Range[145],Complement[Divisors[#],coll[#]]!={}&] (* Jayanta Basu, May 27 2013 *)

A320538 Assuming the truth of the Collatz conjecture, a(n) is the number of divisors of n appearing in the Collatz trajectory of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 2, 4, 2, 6, 2, 4, 3, 5, 2, 4, 2, 6, 2, 4, 2, 8, 3, 4, 2, 6, 2, 6, 2, 6, 3, 4, 3, 6, 2, 4, 3, 8, 2, 4, 2, 6, 3, 4, 2, 10, 3, 6, 3, 6, 2, 4, 3, 8, 2, 4, 2, 9, 2, 4, 2, 7, 4, 6, 2, 6, 2, 6, 2, 8, 2, 4, 2, 6, 3, 6, 2, 10, 2, 4, 2, 6, 2, 4, 2

Offset: 1

Michel Lagneau, Oct 15 2018

nonn

a(p) = 2 for p prime.
a((2^2k - 1)/3) = 2, k = 1, 2, ...
We observe that a(n) differs from A093640(n) for n = 25, 27, 33, 35, 45, 49, 50, 54, 55, 57, 63, 65, 66, 70, 75, 77, 85, ...
7 occurs only eighteen times among the first 65537 terms. - Antti Karttunen, May 18 2019

			a(6) = 4 because the Collatz trajectory 6 -> 3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1 contains 4 divisors of 6: 1, 2, 3 and 6.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006370, A027750, A070165, A093640, A207674, A207675.

lst={}; coll[n_]:=NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&]; Do[AppendTo[lst,Length[Intersection[Divisors[n],coll[n]]]],{n,1,100}]; lst

f(n) = if(n%2, 3*n+1, n/2);
a(n) = {my(kn = n, nb = 1); while (n != 1, n = f(n); if ((kn % n) == 0, nb++);); nb;}

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

Original entry on oeis.org

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

Michel Lagneau, Aug 30 2019

nonn

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.

			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

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

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:

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

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A207675 Numbers such that not all divisors occur in their Collatz trajectories.

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A320538 Assuming the truth of the Collatz conjecture, a(n) is the number of divisors of n appearing in the Collatz trajectory of n.

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

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