cp's OEIS Frontend

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

%I A108117 #19 Dec 23 2024 02:34:07
%S A108117 3494,60674,75494,1122584,2136044,2473934,3367244,5600384,6629804,
%T A108117 6910784,7554644,8572904,10079144,11848094,11892164,12043214,12167594,
%U A108117 12269234,12507284,12700154,13459664,13924544,14495354,15005954,16890914,17827094,20642984,25796054
%N A108117 Numbers n such that prime(k)*n+prime(k+1), for k=1,...,7 all are primes.
%C A108117 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.
%C A108117 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
%e A108117 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 A108117 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 A108117 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 *)
%o A108117 (PARI) isok(n,upto=7)=for(k=1,upto,if(!isprime(prime(k)*n+prime(k+1)),return(0)));1
%o A108117 for(n=1,3*10^7,if(isok(n),print1(n", "))) \\ _Jason Yuen_, Sep 02 2024
%Y A108117 Cf. A067076 (k=1), A088879 (k=2), A111224 (k=3), A101123 (k=4), A102721 (k=5), A108976 (k=7).
%Y A108117 Cf. A108110 (k=1..6), A379427 (k=1..8).
%K A108117 nonn
%O A108117 1,1
%A A108117 _Zak Seidov_, Jun 03 2005
%E A108117 a(13)-a(28) from _Jason Yuen_, Sep 02 2024