%N Primes which are sum of the first k consecutive composite numbers and such that the sums of the first k+1, k+2 and k+3 consecutive composites are also prime.
%C Since there are not 3 consecutive twin primes, with the exception of (3, 5, 7) because one of them would be divisible by 3; there cannot be primes sum of the first consecutive k composites, and such that the sums of the first k+1, k+2, k+3, and k+4 consecutive composites are also prime.
%e a(1) = 1041973527756883 is the least prime, sum of the first consecutive 44208182 composites from 4 to 47042265, and such that the sum of the first 44208183, 44208184 and 44208185 composites are 1041973574799149, 1041973621841417, 1041973668883687, all prime.
%o (PARI) i=0; b=0; for( a=2, 2*10^8, if( !isprime(a) , i=i+1; b=b+a; if(( isprime(b) &a%2==1& isprime(b+a+1)) &isprime(a+2)& isprime(b+2*a+4)&isprime( a+4)&isprime(b+3*a+9), print1(b,", "))))
%Y Cf. A053782, A234847, A235165.
%A _Robin Garcia_, Jan 04 2014
%E a(8)-a(13) from _Donovan Johnson_, Jan 06 2014