Fortunate Prime

Consider the Euclid numbers defined by

where  is the th prime and  is the primorial. The first few values of  are 3, 7, 31, 211, 2311, 30031, 510511, ... (OEIS A006862).

Now let  be the next prime (i.e., the smallest prime greater than ),

where  is the prime counting function. The first few values of  are 5, 11, 37, 223, 2333, 30047, 510529, ... (OEIS A035345).

Then R. F. Fortune conjectured that  is prime for all . The first values of  are 3, 5, 7, 13, 23, 17, 19, 23, ... (OEIS A005235), and values of  up to  are indeed prime (Guy 1994), a result extended to 1000 by E. W. Weisstein (Nov. 17, 2003). The indices of these primes are 2, 3, 4, 6, 9, 7, 8, 9, 12, 18, .... In numerical order with duplicates removed, the Fortunate primes are 3, 5, 7, 13, 17, 19, 23, 37, 47, 59, 61, 67, 71, 79, 89, ... (OEIS A046066).

REFERENCES:

Banderier, C. "Fortunate and Unfortunate Primes: Nearest Primes from a Prime Factorial." Dec. 18, 2000. https://algo.inria.fr/banderier/Computations/prime_factorial.html.

Gardner, M. "Patterns in Primes are a Clue to the Strong Law of Small Numbers." Sci. Amer. 243, 18-28, Dec. 1980.

Golomb, S. W. "The Evidence for Fortune's Conjecture." Math. Mag. 54, 209-210, 1981.

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 7, 1994.

Sloane, N. J. A. Sequences A006862/M2698, A005235/M2418, A035345, and A046066 in "The On-Line Encyclopedia of Integer Sequences."