(**Edit**. What I called "pseudoprimes" are known as "Cramér random primes" in the literature,
of which I was unaware.)

Say that a set $S$ of natural numbers is a set of

*pseudoprimes*if they are (a) random in a sense made explicit below, and (b) the number in $S$ less than $n$ grows as $n / \log_e n$. They are, in a sense, random numbers distributed like the primes. To be more specific, $S$ is created by generating random numbers $u_2, u_3, u_4, \ldots$, uniformly distributed in $[0,1]$, and including $n$ in $S$ iff $u_n < 1 / \log_e n$. (This method was suggested by two MSE users in response to an MSE question on prime-like distributions.) Here are the first twenty numbers in various pseudoprime sets generated by this process: $$3, 8, 10, 11, 12, 13, 15, 28, 35, 39, 44, 50, 53, 65, 66, 67, 68, 70, 78, 86$$ $$8, 27, 29, 31, 32, 33, 41, 50, 52, 55, 57, 62, 63, 71, 72, 74, 78, 82, 83, 92$$ $$5, 28, 31, 34, 44, 45, 46, 48, 50, 53, 59, 64, 68, 69, 70, 72, 76, 78, 85, 95$$ One can imagine other ways of defining pseudoprimes, but let me stick to this simple defintion. Here is their cumulative distribution up to $n=10^6$:

My question is:

Which of the many unsolved problems concerning primes could be proved for pseudoprimes?

For example:

Are there almost surely an infinite number of twin pseudoprimes in any pseuodprime set $S$? (See the graph below for the cumulative distribution.)

Are there almost surely an infinite number of Sophie Germain pseudoprimes?

Can you walk to infinity via a pseudoprime Ulam-spiral? (This was my original motivation.)

Do the Hardy–Littlewood conjectures hold for pseudoprimes?

One can extend this list indefinitely. I am seeking to understand to what extent the $1/\log n$ distribution determines the properties of the primes. Thanks for insights!

will hold. If they would not, they'd been discarded as implausible. (Sorry I am short on time; I might say something more detailed later.) $\endgroup$ – user9072 Jul 26 '12 at 13:08