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The sequence of "prime numbers whose digits are prime": 2, 3, 5, 7, 23, 37, 53, 73, 223, ...

Reference to this sequence: A019546: Primes whose digits are prime

Given an integer $n$, how can I find out the number of numbers appearing in this sequence under $n$? (or, is it possible?)

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    Brute force seems like a useful approach here. Considering we don't even have a way that doesn't rely on brute force somehow to count the number of *primes* less than $n$, I would expect counting the number of these special primes to be just as much of a challenge. – 2017-02-24
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    Brute force accomplishes this in time $O(n^{0.60206})$. That's actually significantly worse than the best algorithms for computing $\pi(n)$, but I can't imagine a way to do better than brute force by more than a power of $\log n$, say. – 2017-02-24

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