cp's OEIS Frontend

(Stems from a post at mymathforum.com)

Outstanding questions include:

(1) Is every term a (prime) square?

(2) How many more terms are there?

The primorial associated with each term hopefully is self-evident. I had to use the "ispseudoprime" function at the higher end. After the first dozen members, c > 0.99 times the defining limit prime(n+1)*prime(n+2). I believe there is a lone entry per primorial.

From Jon E. Schoenfield, Sep 13 2017: (Start)

There is at most one entry per primorial, and every term is the square of a prime. Proof: By definition, since c is a composite, it is the product of at least two primes (not necessarily distinct), and Q (the n-th primorial) is the product of all primes up through prime(n). If c is divisible by any prime <= prime(n), then so is Q + c, so Q + c cannot be prime. The only way that c can be a composite < prime(n+1)*prime(n+2) without being divisible by any prime <= prime(n) is to have c = prime(n+1)^2.

Thus, this sequence simply consists of numbers c of the form prime(k+1)^2 such that primorial(k) + c is prime. (Although k=0 would yield primorial(k) + prime(k+1)^2 = primorial(0) + prime(0+1)^2 = 1 + 2^2 = 5, which is prime, c would be 4, hence not an odd composite.)

Terms of the sequence correspond to the following positive values of k (which are the ones for which primorial(k) + prime(k+1)^2 is prime): 1, 2, 3, 4, 6, 13, 19, 28, 41, 66, 85, 371, 437, 726, 924, 1063, ... (End)