A213329 Smallest k such that there are n - 1 primes between k*p(n) and k*p(n + 1) where p(n) is the n-th prime, or 0 if no such k exists.

1, 2, 2, 3, 8, 5, 13, 9, 8, 25, 10, 15, 31, 19, 19, 15, 56, 0, 33, 79, 26, 33, 0, 21, 54, 110, 52, 126, 57, 16, 71, 42, 140, 29, 130, 0, 51, 76, 51, 53, 179, 0, 192, 93, 216, 34, 34, 107, 247, 120, 84, 278, 0, 84, 105, 99, 301, 95, 154, 287, 0, 40, 154, 325

Offset: 1

Author

Irina Gerasimova, Mar 04 2013

Keywords

• nonn

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,...

Examples

			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.
		

Crossrefs

Cf. 2-gap primes A080192, 3-gap primes A195270.

Programs

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

Extensions

a(13)-a(17) from Charles R Greathouse IV, Mar 06 2013

a(18)-a(64) from Michael B. Porter, Mar 12 2013