A257378 Smallest odd number k such that k*n*2^n+1 is a prime number.

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, 3, 21, 15, 7, 35, 89, 25, 15, 25, 49, 53, 45, 13, 15, 21, 31, 27, 3, 9, 33, 37, 23, 41, 41, 19, 9, 111, 7, 3, 89, 13, 39, 31, 17, 11, 101, 17, 37, 7, 51, 75

Offset: 1

Author

Pierre CAMI, Apr 21 2015

Keywords

• nonn

Comments

Conjecture: a(n) exists for every n.

The conjecture is a corollary of Dirichlet's theorem on primes in arithmetic progressions. - Robert Israel, Jan 05 2016

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

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

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.

Examples

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

Links

• Pierre CAMI, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A256787, A257379, A266909.

Programs

Maple

Q:= proc(m) local k;
  for k from 1 by 2 do if isprime(k*m+1) then return k fi od
end proc: seq(Q(n*2^n), n=1..100); # Robert Israel, Jan 05 2016

Mathematica

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

PARI

a(n) = k=1; while(!isprime(k*n*2^n+1), k+=2); k \\ Colin Barker, Apr 21 2015

PFGW

ABC2 $b*$a*2^$a+1 // {number_primes,$b,1}
a: from 1 to 10000
b: from 1 to 100000 step 2
Charles R Greathouse IV, Apr 24 2015