A035050 a(n) is the smallest k such that k*2^n + 1 is prime.

1, 1, 1, 2, 1, 3, 3, 2, 1, 15, 12, 6, 3, 5, 4, 2, 1, 6, 3, 11, 7, 11, 25, 20, 10, 5, 7, 15, 12, 6, 3, 35, 18, 9, 12, 6, 3, 15, 10, 5, 6, 3, 9, 9, 15, 35, 19, 27, 15, 14, 7, 14, 7, 20, 10, 5, 27, 29, 54, 27, 31, 36, 18, 9, 12, 6, 3, 9, 31, 23, 39, 39, 40, 20, 10, 5, 58, 29, 15, 36, 18, 9, 13

Offset: 0

Author

Labos Elemer

Keywords

• nonn

Comments

From Ulrich Krug (leuchtfeuer37(AT)gmx.de), Jun 05 2010: (Start)

If a(i) = 2 * m then a(i+1) = m.

Proof: (I) a(i) = 2*m, 2*m * 2^i + 1 = m*2^(i+1) + 1 prime, so a(i+1) <= m;

(II) if a(i+1) = m-d for an integer d > 0, (m-d) * 2^(i+1) + 1 = (2*m-2*d) * 2^i + 1 prime;

(2m-2d) < 2m contradiction to a(i) = 2 * m, d = 0.

(End)

Conjecture: for n > 0, a(n) = k < 2^n, so k*2^n + 1 is a Proth prime A080076. - Thomas Ordowski, Apr 13 2019

Examples

			a(3)=2 because 1*2^3 + 1 = 9 is composite, 2*2^3 + 1 = 17 is prime.
a(99)=219 because 2^99k + 1 is not prime for k=1,2,...,218. The first term which is not a composite number of this arithmetic progression is 2^99*219 + 1.
		

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• Gareth A. Jones and Alexander K. Zvonkin, Groups of prime degree and the Bateman-Horn conjecture, 2021.
• Index entries for sequences related to primes in arithmetic progressions

Crossrefs

Analogous case is A034693. Special subscripts (n's for a(n)=1) are the exponents of known Fermat primes: A000215. See also Fermat numbers A000051.

Cf. A007522, A057778, A080076, A085427, A087522, A126717, A127575, A127576, A127577, A127578, A127580, A127581, A127586.

Programs

Magma

sol:=[];m:=1; for n in [0..82] do k:=0; while not IsPrime(k*2^n+1) do k:=k+1; end while; sol[m]:=k; m:=m+1; end for; sol; // Marius A. Burtea, Jun 05 2019

Mathematica

a = {}; Do[k = 0; While[ ! PrimeQ[k 2^n + 1], k++ ]; AppendTo[a, k], {n, 0, 100}]; a (* Artur Jasinski *)
Table[Module[{k=1,n2=2^n},While[!PrimeQ[k*n2+1],k++];k],{n,0,90}] (* Harvey P. Dale, May 25 2024 *)

PARI

a(n) = {my(k = 1); while (! isprime(2^n*k+1), k++); k;}

Formula

a(n) << 19^n by Xylouris' improvement to Linnik's theorem. - Charles R Greathouse IV, Dec 10 2013