I made some numerical simulations about the number of primes which are the sum of 3 consecutive primes (OEIS A034962), that is for instance: $$5+7+11=23$$ $$7+11+13=31$$ $$11+13+17=41$$ $$17+19+23=59$$ $$19+23+29=71$$ $$23+29+31=83$$ $$29+31+37=97$$ $$...$$ The number of such triplets, till a certain integer n, seems to be well approximated by the following function: $$\frac{e\cdot\pi(n)}{\log(n)}$$

If the previous was true, the density of such primes in the whole set of prime numbers would be comparable to that of primes inside the set of naturals.

I ask if this estimate is correct and some references about this matter.

Thanks.