This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A213329 #40 Apr 02 2013 13:09:28 %S A213329 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, %T A213329 52,126,57,16,71,42,140,29,130,0,51,76,51,53,179,0,192,93,216,34,34, %U A213329 107,247,120,84,278,0,84,105,99,301,95,154,287,0,40,154,325 %N 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. %C A213329 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,... %C A213329 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,... %C A213329 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,... %C A213329 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,... %C A213329 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,... %C A213329 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,... %e A213329 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. %o A213329 (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 %Y A213329 Cf. 2-gap primes A080192, 3-gap primes A195270. %K A213329 nonn %O A213329 1,2 %A A213329 _Irina Gerasimova_, Mar 04 2013 %E A213329 a(13)-a(17) from _Charles R Greathouse IV_, Mar 06 2013 %E A213329 a(18)-a(64) from _Michael B. Porter_, Mar 12 2013