I am curious to know if the following number is irrational or transcendental:

$$\displaystyle A = \sum_p 2^{-p},$$ where the sum is over all positive primes. A similar question can be asked for any number $k$ other than 2.

Edit: In retrospect and as pointed out by some comments below that it is trivial that $A$ is irrational. The transcendence question seems still relevant.

If we define a random number

$$\displaystyle b = \sum_n 2^{\rho(n)},$$

where $\rho(n)$ is a Bernoulli random variable defined to be $\rho(n) = -n$ with probability $1/\log n$ and 0 $-\infty$ with probability $1 - 1/\log n$ (so that $2^{\rho(n)} = 0$), for say $n \geq 2$ and define $\rho(1) = -1$.

Can it be proved that say $n$ is transcendental with probability 1?

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Are these numbers irrational and/or transcendental?

I am curious to know if the following number is irrational or transcendental:

$$\displaystyle A = \sum_p 2^{-p},$$ where the sum is over all positive primes. A similar question can be asked for any number $k$ other than 2.

If we define a random number

$$\displaystyle b = \sum_n 2^{\rho(n)},$$

where $\rho(n)$ is a Bernoulli random variable defined to be $\rho(n) = -n$ with probability $1/\log n$ and 0 with probability $1 - 1/\log n$, for say $n \geq 2$ and define $\rho(1) = -1$.

Can it be proved that say $n$ is transcendental with probability 1?