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The Collatz $3x+1$ conjecture claims that any positive integer can be eventually reduced to 1 by iterative application of the maps $x\mapsto 3x+1$ whenever $x$ is odd and $x\mapsto x/2$ whenever $x$ is even.

While the Collatz conjecture is still open, I wonder if the following its relaxed version is any simpler. In this relaxed version we are allowed to apply maps in any order keeping the numbers integer. That is, if $x$ is odd, we still have to apply the $x\mapsto 3x+1$ map; but for even $x$, we have a freedom to choose between $x\mapsto 3x+1$ and $x\mapsto x/2$. The conjecture is that for any positive integer, we can reduce it to 1 with some iterative sequence of maps.

Clearly, the Collatz cojecture would imply the relaxed version. But it may happen that the relaxed version is much simpler. Is it?

The question is inspired by discussion of the sequence http://oeis.org/A109732 which is a permutation of the odd positive integers iff the relaxed version of the $3x-1$ variant of the Collatz conjecture holds.

UPDATE. The minimum number of iterations to reach 1 in this relaxed version is given by http://oeis.org/A127885 and this number is often smaller than that for the Collatz conjecture given by http://oeis.org/A006577 .

The Collatz $3x+1$ conjecture claims that any positive integer can eventually be reduced to $1$ by iterative application of the maps $x \mapsto 3x+1$ whenever $x$ is odd and $x \mapsto x/2$ whenever $x$ is even.

While the Collatz conjecture is still open, I wonder if the following relaxed version is any simpler. In this relaxed version we are allowed to apply maps in any order keeping the numbers integer. That is, if $x$ is odd, we still have to apply the $x \mapsto 3x+1$ map; but for even $x$, we have the freedom to choose between $x \mapsto 3x+1$ and $x \mapsto x/2$. The conjecture is that for any positive integer, we can reduce it to $1$ with some iterative sequence of maps.

Clearly, the Collatz conjecture would imply the relaxed version. But it may happen that the relaxed version is much simpler. Is it?

The question is inspired by discussion of the sequence http://oeis.org/A109732 which is a permutation of the odd positive integers iff the relaxed version of the $3x-1$ variant of the Collatz conjecture holds.

UPDATE. The minimum number of iterations to reach $1$ in this relaxed version is given by http://oeis.org/A127885 and this number is often smaller than that for the Collatz conjecture given by http://oeis.org/A006577 .

The Collatz $3x+1$ conjecture claims that any positive integer can be eventually reduced to 1 by iterative application of the maps $x\mapsto 3x+1$ whenever $x$ is odd and $x\mapsto x/2$ whenever $x$ is even.

While the Collatz conjecture is still open, I wonder if the following its relaxed version is any simpler. In this relaxed version we are allowed to apply maps in any order keeping the numbers integer. That is, if $x$ is odd, we still have to apply the $x\mapsto 3x+1$ map; but for even $x$, we have a freedom to choose between $x\mapsto 3x+1$ and $x\mapsto x/2$. The conjecture is that for any positive integer, we can reduce it to 1 with some iterative sequence of maps.

Clearly, the Collatz cojecture would imply the relaxed version. But it may happen that the relaxed version is much simpler. Is it?

The question is inspired by discussion of the sequence http://oeis.org/A109732 which is a permutation of the odd positive integers iff the relaxed version of the $3x-1$ variant of the Collatz conjecture holds.

UPDATE. The minimum number of iterations to reach 1 in this relaxed version is given by http://oeis.org/A127885 and this number often smaller than that for the Collatz conjecture given by http://oeis.org/A006577 .

The Collatz $3x+1$ conjecture claims that any positive integer can be eventually reduced to 1 by iterative application of the maps $x\mapsto 3x+1$ whenever $x$ is odd and $x\mapsto x/2$ whenever $x$ is even.

While the Collatz conjecture is still open, I wonder if the following its relaxed version is any simpler. In this relaxed version we are allowed to apply maps in any order keeping the numbers integer. That is, if $x$ is odd, we still have to apply the $x\mapsto 3x+1$ map; but for even $x$, we have a freedom to choose between $x\mapsto 3x+1$ and $x\mapsto x/2$. The conjecture is that for any positive integer, we can reduce it to 1 with some iterative sequence of maps.

Clearly, the Collatz cojecture would imply the relaxed version. But it may happen that the relaxed version is much simpler. Is it?

The question is inspired by discussion of the sequence http://oeis.org/A109732 which is a permutation of the odd positive integers iff the relaxed version of the $3x-1$ variant of the Collatz conjecture holds.

UPDATE. The minimum number of iterations to reach 1 in this relaxed version is given by http://oeis.org/A127885 and this number is often smaller than that for the Collatz conjecture given by http://oeis.org/A006577 .

The Collatz $3x+1$ conjecture claims that any positive integer can be eventually reduced to 1 by iterative application of the maps $x\mapsto 3x+1$ whenever $x$ is odd and $x\mapsto x/2$ whenever $x$ is even.

While the Collatz conjecture is still open, I wonder if the following its relaxed version is any simpler. In this relaxed version we are allowed to apply maps in any order keeping the numbers integer. That is, if $x$ is odd, we still have to apply the $x\mapsto 3x+1$ map; but for even $x$, we have a freedom to choose between $x\mapsto 3x+1$ and $x\mapsto x/2$. The conjecture is that for any positive integer, we can reduce it to 1 with some iterative sequence of maps.

Clearly, the Collatz cojecture would imply the relaxed version. But it may happen that the relaxed version is much simpler. Is it?

The question is inspired by discussion of the sequence http://oeis.org/A109732 which is a permutation of the odd positive integers iff the relaxed version of the $3x-1$ variant of the Collatz conjecture holds.

The Collatz $3x+1$ conjecture claims that any positive integer can be eventually reduced to 1 by iterative application of the maps $x\mapsto 3x+1$ whenever $x$ is odd and $x\mapsto x/2$ whenever $x$ is even.

While the Collatz conjecture is still open, I wonder if the following its relaxed version is any simpler. In this relaxed version we are allowed to apply maps in any order keeping the numbers integer. That is, if $x$ is odd, we still have to apply the $x\mapsto 3x+1$ map; but for even $x$, we have a freedom to choose between $x\mapsto 3x+1$ and $x\mapsto x/2$. The conjecture is that for any positive integer, we can reduce it to 1 with some iterative sequence of maps.

Clearly, the Collatz cojecture would imply the relaxed version. But it may happen that the relaxed version is much simpler. Is it?

The question is inspired by discussion of the sequence http://oeis.org/A109732 which is a permutation of the odd positive integers iff the relaxed version of the $3x-1$ variant of the Collatz conjecture holds.

UPDATE. The minimum number of iterations to reach 1 in this relaxed version is given by http://oeis.org/A127885 and this number often smaller than that for the Collatz conjecture given by http://oeis.org/A006577 .