%I
%S 3494,60674,75494,1122584,2136044,2473934,3367244,5600384,6629804,
%T 6910784,7554644,8572904
%N Numbers n such that prime(k)*n+prime(k+1), for k=1,...,7 all are primes.
%C The only n, for which also 19*3494+23 is prime, is n=5600384. In principle, n == 4 (mod 10) can give primes of the form prime(k)*n+prime{k+1), for all k from 1 up to 41, while prime(42)*4+prime(43)=181*4+191 == 5 (mod 10) that is nonprime. It'd be very interesting to find at least one n such that prime[k]*n+prime[k+1), k=1,...,41 are all prime.
%e 3494 is OK because 2*3494+3, 3*3494+5, 5*3494+7, 7*3494+11, 11*3494+13, 13*3494+17 and 17*3494+19 all are primes.
%t s={};Do[If[Union[PrimeQ/@Table[Prime[k]*n+Prime[k+1], {k, 7}]]=={True}, s=Append[s, n]], {n, 4, 10000000, 10}];s
%t Select[Range[9*10^6],AllTrue[Prime[Range[7]]#+Prime[Range[2,8]],PrimeQ]&] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Mar 24 2018 *)
%K nonn
%O 1,1
%A _Zak Seidov_, Jun 03 2005
