A binary irrational with bits defined by primes

Question

Define a number $q$ in binary notation whose $n$-th bit is $1$ for $n$ prime, and $0$ for $n$ composite. So its 2nd, 3rd, 5th, 7th, 11th, etc. bits are $1$, with all other bits $0$. Here is $q$ out to its $101$-st bit: $$.01101010001010001010001000001 010000010001010001000001000 001010000010001010000010001 000001000000010001$$ What is known about $q$? ($\approx 0.414683_{10}$). Has it been investigated? Does it have a name? It is irrational. But is it transcendental?

Answer 1

Your number is known as the prime constant. If we denote it by $\rho$, we have:

$$ \rho =\sum _{{p\in\mathbb{P}}}{\frac {1}{2^{p}}}=\sum _{{n=1}}^{\infty }{\frac {\chi _{{{\mathbb {P}}}}(n)}{2^{n}}}, $$

where $\mathbb{P}$ denotes the set of prime numbers and $\chi _{{{\mathbb {P}}}}$ is the characteristic function of prime numbers, i.e., the function such that for positive integer $n$:

$$ {\displaystyle \chi_\mathbb{P}(n):={\begin{cases}1&{\text{if }}n\in \mathbb{P},\\0&{\text{if }}n\notin \mathbb{P}.\end{cases}}} $$

The decimal expansion of $\rho$ begins with: \begin{align} \rho&=0.414682509851111660248109622\ldots \\ &=0.011010100010100010_2. \end{align}

and is included in the OEIS as sequence A051006.