Here are the numbers whose prime factors all congruent to $\pm 1\pmod 8$:
<http://oeis.org/A058529>


My questions are: 


(1) What is the order of growth of these numbers? That is, what is the order of magnitude of the $n$th smallest among them?


(2) For the numbers whose prime factors are all congruent to $\pm 3\pmod 8$, is the order of growth the same as in (1)?



Growth order of numbers whose prime factors are all congruent to +1 or -1 modulo 8
This is an old question (and according to [this MO question](https://mathoverflow.net/questions/265160/density-of-numbers-whose-prime-factors-belong-to-given-arithmetic-progressions), the result you seek was proven by Landau). In particular, it follows from this that if $S$ is a set of arithmetic progressions containing a density $1/2$ set of the primes, then number of $n \leq x$ all of whose prime factors in $S$ is asymptotic to some constant times $C \frac{x}{\sqrt{\log(x)}}$. This implies that the $n$th positive integer $q\_{n}$ all of whose prime factors are $\pm 1 \pmod{8}$ satisfies $q\_{n} \sim C\_{1} n \sqrt{\log(n)}$. Similarly, if $r\_{n}$ is the $n$th positive integer all of whose prime factors are $\pm 3 \pmod{8}$, then $r\_{n} \sim C\_{2} n \sqrt{\log(n)}$.


If you are looking for a modern treatment of this, which will enable you to compute the constants $C\_{1}$ and $C\_{2}$, this result is stated in [Cojocaru and Murty's book "An introduction to sieve methods and their applications"](https://www.cambridge.org/core/books/an-introduction-to-sieve-methods-and-their-applications/A7B896FAE6B543D73F4FDBDA8609BAC6), but I don't have my copy at home right now, so I can't tell you exactly where in the book to look.


