A379427 -id:A379427 - cp's OEIS frontend

A108110 Numbers n such that prime(k)*n+prime(k+1), for k=1,...,6 all are primes.

284, 3074, 3494, 21698, 32138, 43874, 51794, 60674, 75494, 407348, 437438, 459794, 571478, 660878, 667358, 705464, 716624, 740774, 811028, 820154, 910664, 1059398, 1077998, 1122584, 1150748, 1210754, 1222898, 1265018, 1412174, 1461164, 1486574, 1585868, 1631438

Offset: 1

Zak Seidov, Jun 03 2005

nonn

n == 0 (mod 2). n == 2 (mod 3). n == 3 or 4 (mod 5). - Jason Yuen, Sep 02 2024

			284 is OK because 2*284+3, 3*284+5, 5*284+7, 7*284+11, 11*284+13 and 13*284+17 all are primes.

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

Cf. A108117 (k=1..7), A379427 (k=1..8).

s={};Do[If[Union[PrimeQ/@Table[Prime[k]*n+Prime[k+1], {k, 6}]]=={True}, s=Append[s, n]], {n, 2, 1000000, 2}];s

\\ See isok from A108117
for(n=1,2*10^6,if(isok(n,6),print1(n", "))) \\ Jason Yuen, Sep 02 2024

a(22)-a(33) from Jason Yuen, Sep 02 2024

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

Original entry on oeis.org

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

Zak Seidov, Jun 03 2005

nonn

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

			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.

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).

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 *)

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

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

cp's OEIS Frontend

A108110 Numbers n such that prime(k)*n+prime(k+1), for k=1,...,6 all are primes.

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