A190478 - OEIS

%I #20 Mar 17 2024 02:54:31

%S 2,5,13,3767,19913,726109,4827859,59069473,179993463679,2280987436223

%N a(n) is the smallest prime prime(k) > a(n-1) such that the n numbers 2*prime(j)+3, j=k to k+n-1, are all prime.

%C This essentially searches for blocks of n consecutive primes of the form A023204 (see also A089530) with a minimum of the primes in the block set by the previous entry in the sequence. - _R. J. Mathar_, Jun 02 2011

%C Any further terms are > 10^13. - _Lucas A. Brown_, Mar 17 2024

%H Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/A190478.py">Python program</a>.

%e For n=1, 2 is prime and 2*2+3=7 is prime so a(1)=2.

%e For n=2, 5,7 are consecutive primes 2*5+3 and 2*7+3 are primes so a(2)=5 as 5 is the least such prime > 2.

%e For n=3, 13,17,19 are consecutive primes 2*13+3, 2*17+3, 2*19+3 are primes so a(3)=13 as 13 is the least such prime > 5.

%p isA023204 := proc(n) isprime(n) and isprime(2*n+3) ; end proc:

%p A190478idx := proc(n) option remember; if n = 1 then 1; else for a from procname(n-1)+1 do krun := true; for k from a to a+n-1 do if not isA023204(ithprime(k)) then krun := false; break; end if; end do: if krun then return a; end if; end do: end if; end proc:

%p A190478 := proc(n) ithprime( A190478idx(n)) ; end proc: # _R. J. Mathar_, Jun 02 2011

%o (PARI) old(p,k)=while(k--,p=precprime(p-1));p;

%o n=1;k=0;forprime(p=2,4e9,if(isprime(p<<1+3),if(k++==n,print1(old(p,n)", ");k--;n++),k=0)) \\ _Charles R Greathouse IV_, May 11 2011

%Y Cf. A023204.

%K nonn,more,hard

%O 1,1

%A _Pierre CAMI_, May 11 2011

%E a(8) from _Charles R Greathouse IV_, May 11 2011

%E a(9)-a(10) from _Lucas A. Brown_, Mar 17 2024