Procrastinal problem: Given $n$, one can ask for integers $a,b>1$ of different parities such that $a+b^k$ is prime for $k=(a\bmod 2),\ldots,n$.

A few examples are:

$2+4995825^k$ is prime for $k=0,\ldots,6$.

$1708+6301^k$ is prime for $k=0,\ldots,8$.

$4503+4^k$ is prime for $k=1,\ldots,14$.

Given $n$, do there exist sets of primes of the form $\lbrace a+b^k,k=(a \bmod 2)\ldots,n\rbrace$?

Do there exist such sets with fixed $a\geq 2$, with fixed $b\geq 2$? Given $n$, can one say something on 'smallest' such sets (say with $a+b$ minimal, or with $ab$ minimal)?

Are such sets necessarily finite? (The existence of such an infinite set of primes would be very surprising, see also Density of primes in sequences of the form $a^n+b$)

(Stupid observation: $x^3+1=(x-1)(x^2-x+1)$ implies that $a=1$ is not interesting.)