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A160434 a(n) is the least number k such that (k-th prime after A002110(n)+1) - A002110(n) is not a prime, where A002110(n) is the n-th primorial.

%I A160434 #18 Jun 13 2020 09:58:22
%S A160434 2,3,7,11,20,26,30,37,43,44,42,64,66,46,70,87,99,91,78,95,133,119,113,
%T A160434 133,121,132,134,151,129,204,221,164,176,162,177,169,172,207,234,237,
%U A160434 251,202,231,294,271,298,284,257,254,273,319,267,278,297,309,350,354
%N A160434 a(n) is the least number k such that (k-th prime after A002110(n)+1) - A002110(n) is not a prime, where A002110(n) is the n-th primorial.
%C A160434 The conjecture on the fortunate numbers rephrased with a(n) is a(n)>=2 for all n>=0.
%C A160434 More generally, is a(n) > n+1 always true, or even a(n) > log(n+1)*(n+1)?
%e A160434 a(3)=11: A002110(3)+1=2*3*5+1=31. The 11 primes after 31 are 37, 41, 43, 47, 53, 59, 61, 67, 71, 73 and 79.
%e A160434 Subtracting 2*3*5=30 from each yields: 7, 11, 13, 17, 23, 29, 31, 37, 41, 43, 49.
%e A160434 These are primes except for the 11th value, which is 49=7^2.
%p A160434 a(n):=proc(n) option remember;local k: for k from 1 while isprime((nextprime@@k)(A002110(n)+1)-A002110(n)) do od:
%p A160434 k; end;
%o A160434 (PARI) a(n) = {my(k=0, P=prod(m=1, n, prime(m))); for(m=2, oo, if(ispseudoprime(P+m), k++; if(!isprime(m), return(k)))); } \\ _Jinyuan Wang_, Jun 13 2020
%Y A160434 Cf. A002110, A005235, A160433.
%K A160434 nonn
%O A160434 0,1
%A A160434 Frederick Magata (frederick.magata(AT)web.de), May 13 2009
%E A160434 More terms from _Jinyuan Wang_, Jun 13 2020