I hope that next question is clear here, since my English is bad. In next paragraph I define a sequence of prime numbers as a sum of an even number of prime numbers minus $1$, where previous terms, the primes added, arises from the Erathostenes sieve. I explain my procedure.

Procedure. The first term of our sequence the primes is $3-1$, that is a prime number. The term $3$ arises from the Eratosthenes sieve, being the first prime number that has not been sieved between $2$ and the first integer multiple of $2$ that is $4$ (thus in this step every multiple of $2$ will be sieved).

The second prime that I've defined is $5+7-1$, being the prime numbers $5$ and $7$ the only primes between $3$ and the next multiple that has not been sieved in the Eratosthenes sieve, that is $9$ (and here in this second step every multiple of $3$ will be sieved).

Following previous pattern I find the prime $7+11+13+17+19+23-1=89$, where the meaning of our terms is that are the only primes between our next prime, that is $5$ and the first multiple of $5$ that has not been sieved in previous steps, this is $25$ (here we sieve the multiples of five).

Thus our fourth prime number is $$11+13+19+23+29+31+37+41+43+47-1=293.\square$$

Definition. Thus our sequence of primes $P$, with previous pattern, starts as $2,11,89$ and $293$, following previous pattern, that is prime numbers of the form

$$-1+\sum_{\text{an even quantity}}\text{primes between two consecutive multiples of a prime in previous sieve}.$$

Thus, in previous notation, is required that our terms in previous sum are prime, in an even quantity. I call these integers sieved primes.

Question (Edited, since this question is more reasonable). Can you propose* a conjecture about what should be a sharp (not obvious) upper bound for the counting function of previous sequence of sieved primes $$\pi_{\text{sieved}}(x)=|\{P\leq x,P\text{ is a sieved prime}\}|\leq\text{Upper bound(x)}\tag{1}$$ as $x$ grows over positive reals? Many thanks.

*Thus in this exercise is required that you provide us your computational evidence (provide us more primes of previous kind and their statistics).

I think that this sequence isn't known (I cann't find previous sequence in the OEIS), thus if you want to add some term for the main sequence (the Definition) it also is welcome.

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    $\begingroup$ How do you know there is an even number of primes between $p$ and $p^2$? $\endgroup$ – user5713492 Mar 9 '18 at 12:25
  • $\begingroup$ Yes, I am agree with your doubts about my question, this is the why I've edited my question. Is requred some statistics, computational evidence, to propose a conjecture. Many thanks for your attention @user5713492 $\endgroup$ – user243301 Mar 9 '18 at 12:27

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