Euclid prime

This article defines a property that can be evaluated for a prime number. In other words, every prime number either satisfies this property or does not satisfy this property.
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Definition

A Euclid prime' is a prime number that is one more than a primorial. In other words, it is a prime number of the form $k\#+1$, where $k\#$ denotes the product of the first $k$ primes.

A natural number of the form $k\#+1$ is termed a Euclid number, so a Euclid prime is a Euclid number that happens to be prime.

Occurrence

The ID of the sequence in the Online Encyclopedia of Integer Sequences is A018239

Initial values

The initial values of Euclid numbers are prime. For $k=0,1,2,3,4,5$, the corresponding Euclid numbers $2,3,7,31,211,2311$ are prime. The first Euclid number that is not prime is $30031$, corresponding to $k=6$.