Some smaller PRP/P divisors of "near-cubes":
((2^4-1)^3+2)/11
((2^5-1)^3+2)/3
((2^13-1)^3+2)/3
((2^29-1)^3+2)/3
((2^101-1)^3+2)/3
((2^604-1)^3+2)/11
((2^2733-1)^3+2)/3
((2^7705-1)^3+2)/3
((2^11185-1)^3+2)/3
((2^19905-1)^3+2)/3
((2^23814-1)^3+2)/11
((2^38545-1)^3+2)/3
((2^179294-1)^3+2)/11

To be completed...