Prime related questions

[Jonathan Post]

I personally like, with caveats: "n such that 10^n + prime(n) is prime."

(1) 10^n is arbitrary, albeit common use in OEIS.  One also has many
with 2^n, [e^n], and the like.

(2) Primes are considered in many ways the most interesting integers.
However, they are a special case of the supersequence primes,
semiprimes, 3-almost primes, and so forth.

(3) They are complemented by nonprime(n) and composite(n). They are
iterated by prime(prime(n)), prime(prime(prime(n))) and so forth.

10^n + nonprime(n) is prime: 1, [nor more through 50]

10^n + composite(n) is prime: 4, 49 (since 10^49 + 69 is prime)...

10^n + semiprime(n) is prime: 3, ...

Do you really want to open the doors to these?

One can recover some generality by consider:

Array A[k,n] = n such that k^n + prime(n)

Array A[k,n] = n such that 10^n + prime(n) has exactly k prime factors
(with multiplicity).

And so on. You just have to keep writing those surreal science fiction
books in which such numbers are embedded.