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).
Banderier, C. "Fortunate and Unfortunate Primes: Nearest Primes from a Prime Factorial." Dec. 18, 2000. http://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."
is the 
th
prime and 
is the primorial. The first
few values of 
are 3, 7, 31, 211, 2311, 30031, 510511, ... (OEIS A006862).
be the next prime (i.e., the smallest prime
greater than 
),
is the prime counting function. The first
few values of 
are 5, 11, 37, 223, 2333, 30047, 510529, ... (OEIS A035345).
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).
