OFFSET
1,2
COMMENTS
Smallest prime q such that there is a prime number of primes between q*p(n) and q*p(n + 1) where p(n) is the n-th prime: 5, 3, 2, 2, 5, 2, 7, 2, 2, 2, 2, 13, 13, 3, 2, 2, 3, 2, 2, 7, 2, 3, 2, 3, 2, 7, 3, 7, 3, 3, 3, 2, 7, 2, 7, 2, 3, 3, 3, 2, 11, 2, 11, 5, 29, 3, 7, 3, 7, 2, 3, 11, 2, 2, 2, 5, 3,...
Smallest m such that there are m primes between k*p(n) and k*p(n + 1) for some k > 1 where p(n) is the n-th prime: 1, 1, 1, 2, 1, 2, 1, 2, 0, 3, 1, 1, 1, 3, 3, 0, 2, 2, 0, 3, 1, 2, 4, 1, 0, 1,...
Primes p for which there are no primes between k*p and k*q for some k > 1 where q is the next prime after p: 29, 59, 71, 101,...
Only-one-gap primes: primes p for which there are primes between k*p and k*q for all k > 1 where q is the next prime after p: 2, 3, 5, 7, 11, 13, 17, 19, 23, 31, 37, 41, 43, 47, 53, 61, 67, 73, 79, 83, 89, 97, 103,...
Smallest k such that there is exactly one twin prime pair and no other primes between k*p and k*(p+2) where (p, p+2) is the n-th twin prime pair, or 0 if no such k exists; 3, 2, 5, 4, 2, 10, 2, 6, 0, 3, 0, 7, 0, 6,...
Primes p(n) for which there is exactly one prime quadruplet q, q+2, q+6, q+8 and no other primes between k*p(n) and k*p(n+1) for some k: 61, 163, 197, 271, 409,...
EXAMPLE
For n=4, p(4) = 7 and p(4 + 1) = 11. We are looking for an interval containing 4 - 1 = 3 primes. There are zero primes between 1 * 7 = 7 and 1 * 11 = 11. There are two primes between 2 * 7 = 14 and 2 * 11 = 22 (17 and 19). There are three primes between 3 * 7 = 21 and 3 * 11 = 33 (23, 29, and 31). So a(4) = 3.
PROG
(PARI) a(n)=my(p=prime(n), q=nextprime(p+1), k, t=if(q/p>(1.+1/16597)^(n-1), 2010760, max(exp(1/25/((q/p)^(1./(n-1))-1)), 396738))); while(sum(i=k++*p+1, k*q-1, isprime(i))!=n-1, if(k>t, return(0))); k \\ Charles R Greathouse IV, Mar 06 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
Irina Gerasimova, Mar 04 2013
EXTENSIONS
a(13)-a(17) from Charles R Greathouse IV, Mar 06 2013
a(18)-a(64) from Michael B. Porter, Mar 12 2013
STATUS
approved