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

1,1

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

Links

Table of n, a(n) for n=1..28.

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

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

Cf. A108110 (k=1..6).

Sequence in context: A204960 A249525 A252049 * A233992 A203845 A257316

Adjacent sequences: A108114 A108115 A108116 * A108118 A108119 A108120

Keyword

nonn

Author

Zak Seidov, Jun 03 2005

Extensions

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

Status

approved