Consider $\lfloor 10^n \times 0.731926765612646213686753345587262244668218433356357832021 \rfloor$. For each $n$, reverse the digits. If the number isn't already prime, find the next prime.

For the first 24 values of $n$, the result is prime. This start is based on the record left-truncatable prime 357686312646216567629137.

After that, one must add the following values to get a prime number $(12, 0, 16, 2, 12, 20, 2, 2, 0, 14, 0, 2, 12, 6, 6, 10, 10, 10, 16, 12, 10, 4, 14, 2, 10, 12, 2, 0, 2)$

All values bounded above by 20. If this could go on (it doesn't), it would provide a solution for the finding primes Polymath project. How far can something like this be extended for a given upper bound? For $n$ going from 1 to 100, what constant minimizes the farthest distance to the corresponding 1 to 100 digit prime?

A similar question -- what constants work best without the reverse-digits step? For upper bound 0, $\lfloor 10^n \times .73939133 \rfloor$ could be used to produce primes for $n=1..8$. How far could this be extended with an upper bound of 10?

Is the following provable?

For any $0<k<1$ and any positive integer $d$ there exists an $n$ such that the following values are composite.

$$\lfloor 10^n \times k \rfloor, \lfloor 10^n \times k \rfloor+1, ..., \lfloor 10^n \times k \rfloor +d$$

Added results.

Prime to $n=31$,
$\lfloor 10^n \times .19992191983120858559939829 \rfloor$ + $(1, 0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 0, 2, 0, 2, 2, 1, 1, 2)_n$

Prime to $n=43$, $\lfloor 10^n \times .2249916676947644237216638216097398882368887 \rfloor$ + $(0, 1, 3, 2, 2, 2, 1, 1, 0, 1, 0, 3, 2, 1, 3, 3, 1, 0, 2, 1, 0, 3, 3, 0, 1, 1, 0, 1, 1, 0, 2, 0, 0, 3, 1, 3, 1, 0, 1, 1, 3, 1, 0)_n$

Reverse prime to $n=50$, $\lfloor 10^n \times .79946519968197358169933425438229277782582723528486 \rfloor$ + $(0,0,0,2,0,0,2,0,0,0,0,2,2,2,2,2,0,0,0,0,0,0,0,2,2,0,0,2,0,2,2,2,0,2,2,0,2,0,0,2,0,2,2,2,0,2,2,2,0,0)_n$

Reverse prime to $n=64$, $\lfloor 10^n \times .1319788512835399616489432175746323115633163831863127383662373426 \rfloor$ + $(1,0,0,2,2,2,0,2,0,2,2,2,0,0,0,0,0,2,0,2,0,2,2,2,2,2,0,2,0,2,0,0,0,0,0,8,8,6,8,6,2,8,8,0,8,2,0,6,8,2,8,2,8,0,6,6,0,0,6,8,6,8,0,2)_n$