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

1,2

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

Links

Table of n, a(n) for n=1..35.

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

Example

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.

Mathematica

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

Prog

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

Crossrefs

Cf. A080076, A331539, A331540.

Sequence in context: A055391 A177940 A304590 * A019612 A007444 A332309

Adjacent sequences: A134873 A134874 A134875 * A134877 A134878 A134879

Keyword

nonn

Author

T. D. Noe, Nov 17 2007

Extensions

More terms from Charles R Greathouse IV, Mar 18 2010

Status

approved