A108117 Numbers n such that prime(k)*n+prime(k+1), for k=1,...,7 all are primes.

3494, 60674, 75494, 1122584, 2136044, 2473934, 3367244, 5600384, 6629804, 6910784, 7554644, 8572904, 10079144, 11848094, 11892164, 12043214, 12167594, 12269234, 12507284, 12700154, 13459664, 13924544, 14495354, 15005954, 16890914, 17827094, 20642984, 25796054

Offset: 1

Author

Zak Seidov, Jun 03 2005

Keywords

• nonn

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

Examples

			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.
		

Crossrefs

Cf. A067076 (k=1), A088879 (k=2), A111224 (k=3), A101123 (k=4), A102721 (k=5), A108976 (k=7).

Cf. A108110 (k=1..6), A379427 (k=1..8).

Programs

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 *)

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

Extensions

a(13)-a(28) from Jason Yuen, Sep 02 2024