Irrationality of 0.123456789101112 ... and similar numbers

Question

Consider four numbers in $(0,1)$: $n_1$ in base $10$ is formed by listing the decimal digits $1,2,3,4,\ldots$; $b_1$ in binary is formed by $0$ and $1$ for each even and odd digit of $n_1$: $$ n_1 = 0.123456789101112131415161718192021 \ldots $$ $$ b_1 = 0.101010101101110111011101110110001 \ldots $$ $n_2$ and $b_2$ are formed similarly, but listing the primes $2,3,5,7,\ldots$: $$ n_2 = 0.23571113171923293137414347535961 \ldots $$ $$ b_2 = 0.01111111111101011111010101111101 \ldots $$ Which of these numbers is known to be {rational, irrational, algebraic, transcendental}? I presume that all four are irrational.

Answer 1

$n_1$ is the Champernowne constant, which is known to be normal. $n_2$ is the Copeland–Erdős constant, which is also known to be normal in base $10$.