A225911 - OEIS

%I #28 Sep 08 2022 08:46:05

%S 1,1,2,1,10,3,3,3,12,2,2,15,17,11,3,8,2,10,12,2,73,35,21,11,18,3,12,2,

%T 3,28,48,8,11,31,17,102,17,7,17,8,2,35,13,135,33,72,12,2,18,3,26,17,

%U 38,16,51,12,2,2,2,40,103,45,26,40,16,3,10,26,10,8,2,11

%N Smallest k such that k*6^n+1 is prime.

%C In average k~0.6*n and 0 < k < 8*n until a proof that k may be > 8*n.

%C Dirichlet's theorem proves that a(n) exists for each n. Linnik's theorem gives bounds; in particular the version due to Xylouris gives a(n) << 1855^n. - _Charles R Greathouse IV_, May 20 2013

%H Pierre CAMI, <a href="/A225911/b225911.txt">Table of n, a(n) for n = 1..3000</a>

%H <a href="/index/Pri#primes_AP">Index entries for sequences related to primes in arithmetic progressions</a>

%e 6^1+1=7 is prime, so  a(1)=1;

%e 6^2+1=37 is prime, so a(2)=1;

%e 6^3+1=217 is composite, 2*6^3+1=433 is prime, so a(3)=2.

%o (PFGW & SCRIPTIFY)

%o SCRIPT

%o DIM n,0

%o DIM k

%o DIMS t

%o OPENFILEOUT myf,a(n).txt

%o LABEL a

%o SET n,n+1

%o IF n>3000 THEN END

%o SET k,0

%o LABEL b

%o SET k,k+1

%o SETS t,%d,%d\,;n;k

%o PRP k*6^n+1,t

%o IF ISPRP THEN GOTO c

%o GOTO b

%o LABEL c

%o WRITE myf,t

%o GOTO a

%o (Magma) S:=[]; for n in [1..100] do k:=1; while not IsPrime(k*6^n+1) do k:=k+1; end while; Append(~S, k); end for; S; // _Bruno Berselli_, May 20 2013

%o (PARI) a(n)=my(k);while(!ispseudoprime(k++*6^n+1),);k \\ _Charles R Greathouse IV_, May 20 2013

%Y Cf. A225941, A035050.

%K nonn

%O 1,3

%A _Pierre CAMI_, May 20 2013