Primes in Geometric-Arithmetic Progression

A geometric-arithmetic progression of primes is a set of $k$ primes (denoted by GAP-$k$) of the form $p_1 r^j + j d$ for fixed $p_1$, $r$ and $d$ and consecutive $j$, {\it i.e}, \{p_1, \, p_1 r + d, \, p_1 r^2 + 2 d, \, p_1 r^3 + 3 d, \,...}. We study the conditions under which, for $k \ge 2$, a GAP-$k$ is a set of $k$ primes in geometric-arithmetic progression. Computational data (along with the MATHEMATICA codes) containing progressions up to GAP-13 is presented. Integer sequences for the sets of differences $d$ corresponding to the GAPs of orders up to 11 are also presented.