Consider the subset of odd positive integers defined and constructed as follows by these rules :
A) $1$ is in the set.
B) if $x$ is in the set , then $2x + 1$ is in the set.
C) if $x$ and $y$ are in the set then $xy$ is in the set.
I call them extended Mersenne numbers because rule A and B alone give the Mersenne numbers $2^n - 1$.
And every product of Mersenne numbers must be in the set as well.
So the set or list of extended Mersenne numbers starts like
$$1,3,7,9,15,19,21,27,31,39,43,45,49,55,57,63,79,81,87,91,93,99,...$$
The main question is how dense is this within the odd integers ?
Secondary is there an easy and efficient test to know if an odd integer is in the list ?
I assume this sequence has no name yet but I could be wrong ?
*** speculation ***
I see $22$ odd elements from $1$ till $99$, which is close to the number of odd primes under $99$ (that is $24$) $$ \dfrac11 + \dfrac13 + \dfrac17 + \dfrac19 + \ldots + \ldots+\dfrac{1}{99} = 2.034980..$$
So this (subjectively I admit ) seems to imply the sequence grows quite fast. Therefore I guess maybe an asymptotic of the form
$C* x * \ln(x)^D $
for some constants $C,D$ might be an asymptotic for the counting function.
On the other hand I suspect much sharper asymptotics can be proven.
Background
This comes from abstract algebra where we want a commutative latin square with units and inverses such that "Property A"
$$ x * (y * y^{-1}) = x = (x * y) * y^{-1} $$
holds for all elements $x,y$.
( if this property has a name please tell me )
and for every element $z$
$$ z^2 = 1 $$
( not the main question here, but I wonder how dropping the $z^2 = 1$ condition effects things )
The dimensions of these are then exactly these extended Mersenne numbers + $1$.
edit
corrected forgotten terms and copy lulu's conjecture :
The sequence is equal to https://oeis.org/A197625 ???
( https://math.stackexchange.com/users/252071/lulu )
edit2
Greg Martin found counterexample 219 so lulu's conjecture is false.
Greg also conjectured that $D=0$ or $D$ goes to $0$.
It seems that Greg believes we will get close to linear, but I want to point out that the set is denser than the products of mersenne numbers.
On the other hand numbers like $63 = 3 * 21 = 2*31 + 1$ can be reached in $2$ ways so the rules have overlapping values, which could lower the density.
I would not be surprised if the density is slightly higher than the sum of 2 squares or the density of $a^3 + b^3 + c^3$ which are both less than linear.
It reminds me a bit of collatz, where the rules $2x$ and $(x-1)/3$ generate all integers , or so we believe.
But these rules for the extented Mersenne numbers have no deminishing rule like $(x-1)/3$ ( deminish because that is smaller than $x$) , which is why I believe they might not have linear density.
added
Noticed how powers of $5$ or powers of $17$ are never in the list :
$5$ and $25$ are not in.
So $125,625,…$ are not created By products. Also $125,625,…$ are of the form $4n + 1$, so they did not come from the 2x + 1 rule either.
Similar with $17$ or other primes missing of the form $4n + 1$.
This ofcourse highly influences the density.
Also notice primes of the form $4n+1$ have to come from 2x + 1 themselves , where x must be odd !!
2 x + 1 maps 1 mod 4 to 3 mod 4, and maps 3 mod 4 to 3 mod 4 !!
So primes 1 mod 4 are a key thing here !!