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A190478 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.

%I A190478 #20 Mar 17 2024 02:54:31
%S A190478 2,5,13,3767,19913,726109,4827859,59069473,179993463679,2280987436223
%N A190478 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 A190478 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 A190478 Any further terms are > 10^13. - _Lucas A. Brown_, Mar 17 2024
%H A190478 Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/A190478.py">Python program</a>.
%e A190478 For n=1, 2 is prime and 2*2+3=7 is prime so a(1)=2.
%e A190478 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 A190478 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 A190478 isA023204 := proc(n) isprime(n) and isprime(2*n+3) ; end proc:
%p A190478 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 A190478 := proc(n) ithprime( A190478idx(n)) ; end proc: # _R. J. Mathar_, Jun 02 2011
%o A190478 (PARI) old(p,k)=while(k--,p=precprime(p-1));p;
%o A190478 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 A190478 Cf. A023204.
%K A190478 nonn,more,hard
%O A190478 1,1
%A A190478 _Pierre CAMI_, May 11 2011
%E A190478 a(8) from _Charles R Greathouse IV_, May 11 2011
%E A190478 a(9)-a(10) from _Lucas A. Brown_, Mar 17 2024