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I am looking for sequences $\{a_n\} \subset \mathbb{N}$ with the following properties:

(1) $\displaystyle \# \{a_n \leq x \} = \frac{x}{\sqrt{\log x}} + O_\epsilon(x^{1/2 + \epsilon})$, and

(2) $\# \{a_n : a_n \equiv a \pmod{m}\} = \frac{x}{\phi(m)\sqrt{\log x}} + O_\epsilon(x^{1/2 + \epsilon})$ whenever $\gcd(a,m) = 1$.

In other words, $\{a_n\}$ looks like a thicker version of the primes.

Are there any such examples?

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    $\begingroup$ If the $a_n$ can have repeats, then you should just be able to repeat primes an appropriate number of times. If the $a_n$ are distinct, then there is no such sequence. Perhaps you could clarify the question. $\endgroup$
    – Lucia
    Commented Jan 19, 2015 at 20:19
  • $\begingroup$ @Lucia What is the problem in the case of the $a_n$ distinct? Do you need all the modules $m$ to get the contradiction? $\endgroup$
    – juan
    Commented Jan 19, 2015 at 20:41
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    $\begingroup$ @juan When the $a_n$ are distinct, then from the conditions it follows that very few of the $a_n$ can be multiples of a prime $p$ with $p\le x^{1/10}$ say. But then the number of integers up to $x$ composed of primes larger than $x^{1/10}$ is $O(x/\log x)$. $\endgroup$
    – Lucia
    Commented Jan 19, 2015 at 21:53
  • $\begingroup$ That comment is satisfactory for my purposes, thank you $\endgroup$
    – Stanley Yao Xiao
    Commented Jan 20, 2015 at 17:26

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