%I #28 Feb 18 2016 03:51:40

%S 3,3,9,15,3,3,9,3,15,15,9,3,33,9,81,21,9,3,27,27,33,27,45,45,33,27,15,

%T 33,45,3,39,81,9,75,81,3,15,15,81,27,3,9,9,15,189,27,27,15,105,27,75,

%U 93,51,177,57,27,75,99,27,45,105,105,9,27,9,3,9,237

%N Smallest odd number k divisible by 3 such that k*2^n + 1 is prime.

%C As N increases, (Sum_{n=1..N} a(n))/(Sum_{n=1..N} n) appears to approach 2*log(2), as can be seen by plotting the first 31000 terms.

%C This observation is consistent with the prime number theorem as the probability that k*2^n+1 is prime is 1/(n*log(2)+log(k))/2 for k multiple of 3 so ~ 1/(2*n*log(2)) as n increases, if k ~ 2*n*log(2) then k/(2*n*log(2)) ~ 1.

%H Pierre CAMI, <a href="/A264098/b264098.txt">Table of n, a(n) for n = 1..31000</a>

%e 3*2^1 + 1 = 7 is prime so a(1) = 3.

%e 3*2^2 + 1 = 13 is prime so a(2) = 3.

%e 3*2^3 + 1 = 25 is composite; 9*2^3 + 1 = 73 is prime so a(3) = 9.

%p for n from 1 to 100 do

%p for k from 3 by 6 do

%p if isprime(k*2^n+1) then

%p A[n]:= k; break

%p fi

%p od

%p od:

%p seq(A[n],n=1..100); # _Robert Israel_, Jan 22 2016

%t Table[k = 3; While[! PrimeQ[k 2^n + 1], k += 6]; k, {n, 68}] (* _Michael De Vlieger_, Nov 03 2015 *)

%o (PFGW & SCRIPT)

%o Command: pfgw64 -f -e500000 in.txt

%o in.txt SCRIPT FILE:

%o SCRIPT

%o DIM k

%o DIM n, 0

%o DIMS t

%o OPENFILEOUT myf, a(n).txt

%o LABEL loop1

%o SET n, n+1

%o SET k, -3

%o LABEL loop2

%o SET k, k+6

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

%o PRP k*2^n+1, t

%o IF ISPRP THEN WRITE myf, t

%o IF ISPRP THEN GOTO loop1

%o GOTO loop2

%o (PARI) a(n) = {k = 3; while (!isprime(k*2^n+1), k += 6); k;} \\ _Michel Marcus_, Nov 03 2015

%Y Cf. A035050, A057778, A264097.

%K nonn

%O 1,1

%A _Pierre CAMI_, Nov 03 2015