OFFSET
1,1
COMMENTS
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.
There are no values of n such that prime(k)*n+prime(k+1), k=1,...,9 are all prime. Proof: If n = 3*i then 2*(3*i)+3 = 3*(2*i+1) is not prime. If n = 3*i+1 then 5*(3*i+1)+7 = 3*(5*i+4) is not prime. If n = 3*i+2 then 23*(3*i+2)+29 = 3*(23*i+25) is not prime. - Jason Yuen, Sep 02 2024
EXAMPLE
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.
MATHEMATICA
s={}; Do[If[Union[PrimeQ/@Table[Prime[k]*n+Prime[k+1], {k, 7}]]=={True}, s=Append[s, n]], {n, 4, 10000000, 10}]; s
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 *)
PROG
(PARI) isok(n, upto=7)=for(k=1, upto, if(!isprime(prime(k)*n+prime(k+1)), return(0))); 1
for(n=1, 3*10^7, if(isok(n), print1(n", "))) \\ Jason Yuen, Sep 02 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Zak Seidov, Jun 03 2005
EXTENSIONS
a(13)-a(28) from Jason Yuen, Sep 02 2024
STATUS
approved