TITLE: Is there any number greater than 8 of the form $2^{2k+1}$ which is the sum of a prime and a safe prime?
QUESTION [10 upvotes]: Is there any number greater than 8 of the form $2^{2k+1}$ which is the sum of a prime and a safe prime?  While answering @pedja's question about the existence of any such representations I was surprised to discover that $32$, $128$, $512$, and $2048$ cannot be represented in this way, and I've since found that the same is true for $8192$, $32768$, $131072$, $524288$, $2097152$, and $8388608$.  Is there any counterexample?  If not, how can it be proved?

REPLY [10 votes]: From Nicolas's argument I realized that it was sufficient to check if $2^{2n+1}-7$ was prime. A much easier task than the usual brute forcing. After a few minutes of manual testing on wolfram alpha I found that $2^{39}$ could indeed be written as the sum of a prime and a safeprime:
$$7+549755813881=549755813888=2^{39}.$$
Edit: I wrote a sage program to check all of the odd powers of two less than 2000. It seems I was a bit lucky since the next examples are 715 and 1983. I checked OEIS and numbers such that $2^{n}-7$ are prime has its own page.