Consider these two sets of odd positive integers.
SET 1 :
constructed by these rules :
a) $1$ is in the set.
b) if $x$ is in the set, then so is $2 x + 1$.
c) if $x$ and $y$ are in the set then so is $xy$.
This set or sequence starts like
$$1,3,7,9,15,19,21,27,31,39,43,45,49,55,57,63,79,81,87,91,93,99,...$$
SET 2 :
constructed by these rules :
$1$ is in the set.
if $a$ is in the set, then so is $2 a + 5$.
if $a$ and $b$ are in the set then so is $ab$.
Let $\pi_1(n)$ be the counting function for numbers in the set SET1 up to $n$. Let $\pi_5(n)$ be the counting function for numbers in the set SET2 up to $n$.
If we consider $\mod 10$ (or the last digit) we see that SET1 can have integers of type
$$1,3,5,7,9 \mod 10$$
And likewise SET2 can have integers of type
$$1,3,7,9 \mod 10$$
Indeed SET2 can never have multiples of $5$ by the way it is constructed.
So one set has 5 residu and the other 4.
The growth rate of both sets are an interesting question. And the potential similarities between them.
Some will argue that SET1 has a limiting density of 1 for the odd integers. And/or a limiting density of 1 for all these odd residu mod 10.
Some will probably argue that SET2 grows much faster because $2x + 1 < 2 x + 5$ , while others will expect about the same growth rate.
The questions about SET1 have already been posted and argued here :
Density of extended Mersenne numbers?
https://mathoverflow.net/questions/440747/density-of-extended-mersenne-numbers
So I post this somewhat follow-up for SET2.
I will not ask how fast it grows or what its exact density is by itself. Rather I wonder how it relates to the other set.
QUESTION 1 :
Do we have
$$ \lim_{n \to \infty} \frac{\pi_1(n)}{\pi_5(n)} = \frac{5}{4} $$
??
QUESTION 2 :
Do we ever have
$$\pi_1(m) = \pi_5(m)$$
for $m>2$
??
Plots are welcome too !
added :
For SET2 we have a similar property as for SET1
$2 a + 5$ 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 !!
those primes $1 \mod 4$ cannot be reached.
Added :
If I am not mistaken SET2 starts as
$$1,7,19,43,49,91,103,133,187,211,271,301,343,361,379,..$$
