Here are some number theory questions that (so far as I know) are original with me and appeared on MathOverflow first . They may be hard; at this writing none of them are satisfactorily answered.

Jumping prime sieve: How do these primes jump?

Playing prime leapfrog: Playing leapfrog with primes

Semiprime cover: How much do these interval collections cover?

Combinatorial ring toss: Has anyone seen this version of ring toss (combinatorial object) before?

Square difference sequences: How many sequences of rational squares are there, all of whose differences are also rational squares?

Gerhard "Will Accept Fame From Questions" Paseman, 2017.09.02.