Pick a positive integer to start a sequence. Now double it, and add one to the result: this is the second number of your sequence. Double that number, and add one, and that will be your third number; repeat the doubling and adding, and you will have a fourth number, and so on.
If you start with a prime number, and you keep doubling and adding one, is it possible to produce a sequence with only prime numbers?
Starting with 3 you get: 3, 7, 15... the third number is a multiple of 3.
Start with 5: 5, 11, 23, 47, 95... the fifth number is a multiple of 5.
Start with 7: 7, 15, 31, 63, 127, 255, 511... the seventh number is a multiple of 7.
I'm guessing that the p-th number in each sequence is a multiple of p, the first number, and I'd bet on Fermat's theorem about ašp modulus p.