%I #27 Jan 06 2016 15:52:56

%S 1,5,3,3,13,3,3,9,5,13,9,3,3,5,9,7,3,3,3,5,3,7,19,5,5,33,3,7,7,9,5,15,

%T 3,21,15,7,35,89,25,15,25,49,53,45,13,15,21,31,27,3,9,33,37,23,41,41,

%U 19,9,111,7,3,89,13,39,31,17,11,101,17,37,7,51,75

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

%C Conjecture: a(n) exists for every n.

%C The conjecture is a corollary of Dirichlet's theorem on primes in arithmetic progressions. - _Robert Israel_, Jan 05 2016

%C As N increases sum {k, n=1 to N} / sum {n, n=1 to N} tends to 0.818.

%C If k=1 then n*2^n+1 is a Cullen prime.

%C Generalized Cullen primes have the form n*b^n+1, I propose to name the primes k*n*2^n-1 generalized Cullen primes of the second type.

%H Pierre CAMI, <a href="/A257378/b257378.txt">Table of n, a(n) for n = 1..10000</a>

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

%e 1*2*2^2+1=9 composite, 3*2*2^2+1=25 composite, 5*2*2^2+1=41 prime so a(2)=5.

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

%p Q:= proc(m) local k;

%p for k from 1 by 2 do if isprime(k*m+1) then return k fi od

%p end proc: seq(Q(n*2^n), n=1..100); # _Robert Israel_, Jan 05 2016

%t Table[k = 1; While[!PrimeQ[k*n*2^n + 1], k += 2]; k, {n, 73}] (* _Michael De Vlieger_, Apr 21 2015 *)

%o (PFGW & SCRIPT)

%o SCRIPT

%o DIM n,0

%o DIM k

%o DIMS t

%o OPENFILEOUT myf,a(n).txt

%o LABEL loop1

%o SET n,n+1

%o IF n>3000 THEN END

%o SET k,-1

%o LABEL loop2

%o SET k,k+2

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

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

%o IF ISPRP THEN GOTO a

%o GOTO loop2

%o LABEL a

%o WRITE myf,t

%o GOTO loop1

%o (PARI) a(n) = k=1; while(!isprime(k*n*2^n+1), k+=2); k \\ _Colin Barker_, Apr 21 2015

%o (PFGW) ABC2 $b*$a*2^$a+1 // {number_primes,$b,1}

%o a: from 1 to 10000

%o b: from 1 to 100000 step 2

%o _Charles R Greathouse IV_, Apr 24 2015

%Y Cf. A256787, A257379, A266909.

%K nonn

%O 1,2

%A _Pierre CAMI_, Apr 21 2015