Deductions & credits

see this link. generally, there are no premiums for just part A and if you delay signing up once you're 65 there can be penalties for signing up later

there is nothing improper about being covered by part A Medicare and an HDHP or any othe type of health plan. 

https://www.medicare.gov/basics/get-started-with-medicare/medicare-basics/what-does-medicare-cost#:~...