[seqfan] A039669 and new related numbers

[Tomasz Ordowski]

Dear OEIS Community!

Let's define:
Numbers k such that all positive values of k - 2^(2^m) are prime,
with integer m >= 0. Many Data: 4, 7, 9, 15, 21, 33, 45, 63, 75,
105, 153, 183, 195, 243, 273, 285, 435, 525, 573, 603, 813, 825, ...
If k > 4 is a term of this sequence, then k-2, k-4 is a twin prime pair.
So all terms k > 7 are divisible by 3 and k = 7 is the only prime here.
It seems that there are infinitely many such numbers. Find more...
Note though that the set {A039669} is finite and probably complete.
See https://oeis.org/A039669 (this is a subset of wanted numbers).
Neil, are these my numbers worth a new sequence in the OEIS?
If so, I'll submit my numbers because I already have a free edition.
Max, if you have any criticisms, please express them now, thanks!
Below PS (not only for those interested).

Best regards,

Thomas Ordowski (2/3).

PS. Similarly. Interesting, but with a poor data section, namely:
Numbers k such that all positive values of k - 2^(2^n) are square,
with natural n > 0. Data: {5, 8, 13, 20}. Please "more", if any...
Note that 2^(2^n) for n > 0 is the square (2^(2^(n-1))^2.
If there are no more terms, what do we do with it?
Prove it and forget about the OEIS. Yes?