A134876 Number of Proth primes: number of primes of the form 1 + k*2^n with k odd and k < 2^n.

1, 2, 1, 3, 4, 8, 18, 23, 44, 73, 142, 277, 484, 871, 1644, 3060, 5851, 10917, 20776, 39263, 74752, 142521, 271223, 520242, 996486, 1916486, 3686628, 7103236, 13702428, 26469008, 51193351, 99099882, 192044541, 372559804, 723389144

Offset: 1

Author

T. D. Noe, Nov 17 2007

Keywords

• nonn

Comments

The ratio a(n+1)/a(n) is about 2 * n /(n+1). - Corrected by Thomas Ordowski, Oct 17 2014

Conjecture: a(n) ~ C * 2^n / n, where C = 1/(2 log 2) = 0.7213475... - Thomas Ordowski, Oct 17 2014

Examples

			a(1)=1 because 3 is the only Proth prime for n=1.
a(2)=2 because 5 and 13 are the only primes for n=2.
a(3)=1 because 41 is the only prime for n=3.
		

Links

• Eric Weisstein's World of Mathematics, Proth's Theorem

Crossrefs

Cf. A080076, A331539, A331540.

Programs

Mathematica

Table[cnt=0; Do[If[PrimeQ[1+k*2^n], cnt++ ], {k,1,2^n,2}]; cnt, {n,20}]

PARI

a(n) = my(s=0);forstep(k=1,2^n-1,2,s+=ispseudoprime(k<Jeppe Stig Nielsen, Jan 19 2020

Extensions

More terms from Charles R Greathouse IV, Mar 18 2010