Irrationality of "primes coded in binary" For fun, I have been considering the number
$$
\ell := \sum_{p} \frac{1}{2^p}
$$
It is clear that the sum converges and hence $\ell$ is finite.  $\ell$ also has the binary expansion
$$
\ell = 0.01101010001\dots_2
$$
with a $1$ in the $p^{th}$ place and zeroes elsewhere.  I have also computed a few terms (and with the help of Wolfram Alpha, Plouffe's Inverter, and this link from Plouffe's Inverter) I have found that $\ell$ has the decimal expansion 
$$
\ell = .4146825098511116602481096221543077083657742381379169778682454144\dots.
$$
Based on the decimal expansion and the fact that $\ell$ can be well approximated by rationals, it seems exceedingly likely that $\ell$ is irrational.  However, I have been unable to prove this.
Question: Can anyone provide a proof that $\ell$ is irrational?

That $\ell$ is irrational is clear. There are arbitrarily large gaps between consecutive primes, so the binary expansion of $\ell$ cannot be periodic. Any rational has a periodic binary expansion. 
The fact that there are arbitrarily large gaps between consecutive primes comes from observing that if $n>1$, then all of $n!+2, n!+3, \dots, n!+n$ are composite.
