1,1

The only n, for which also 19*n+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

Robert Israel, Table of n, a(n) for n = 1..500

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.

filter:= proc(n) andmap(isprime, [2*n+3, 3*n+5, 5*n+7, 7*n+11, 11*n+13, 13*n+17, 17*n+19]) end proc:

select(filter, [seq(14+30*t, t=0..10^6)]); # Robert Israel, Apr 06 2026

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

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).

Sequence in context: A204960 A249525 A252049 * A233992 A203845 A257316

Adjacent sequences: A108114 A108115 A108116 * A108118 A108119 A108120

nonn,changed

Zak Seidov, Jun 03 2005

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

approved