A084957 - OEIS

%I #20 Feb 04 2020 06:35:11

%S 2,2,2,2,1447,9769,17231,17231,32611,18527009,161205841,3123824801,

%T 26813406071,4398156030379,4398156030379

%N Initial prime of the first prime chain of length n under the iteration x -> 4x + 3.

%C This is a special case of prime chains generated by f(x) = c*x + d.

%H D. H. Lehmer, <a href="https://doi.org/10.1112/plms/s3-14A.1.183">On certain chains of primes</a>, Proc. London Math. Soc. (3) 14a 1965 183-186.

%e a(3) = 2 since 2, f(2) = 11, and f(11) = 47 are primes when f(x) = 4*x + 3.

%t c[p_] := Block[{k=1, q=4*p+3}, While[ PrimeQ[q], q=4*q+3; k++]; k]; a[n_] := Block[ {p=2}, While[c[p] < n, p = NextPrime@ p]; p]; Array[a, 9] (* _Giovanni Resta_, Mar 21 2017 *)

%o (PARI) has(p,n)=for(i=2,n, if(!isprime(p=4*p+3), return(0))); 1

%o a(n)=forprime(p=2,, if(has(p,n), return(p))) \\ _Charles R Greathouse IV_, Jan 20 2017

%Y Cf. A057330, A057331, A083388, A084954, A084955, A084956, A084958, A084959, A084960, A084961.

%K nonn,more

%O 1,1

%A _W. Edwin Clark_, Jun 14 2003

%E a(11)-a(12) from _Donovan Johnson_, Sep 27 2008

%E a(13) from _John Cerkan_, Jan 20 2017

%E a(14)-a(15) from _Giovanni Resta_, Mar 21 2017