A134876 - OEIS

%I #20 Feb 18 2020 02:06:43

%S 1,2,1,3,4,8,18,23,44,73,142,277,484,871,1644,3060,5851,10917,20776,

%T 39263,74752,142521,271223,520242,996486,1916486,3686628,7103236,

%U 13702428,26469008,51193351,99099882,192044541,372559804,723389144

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

%C The ratio a(n+1)/a(n) is about 2 * n /(n+1). - Corrected by _Thomas Ordowski_, Oct 17 2014

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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ProthsTheorem.html">Proth's Theorem</a>

%e a(1)=1 because 3 is the only Proth prime for n=1.

%e a(2)=2 because 5 and 13 are the only primes for n=2.

%e a(3)=1 because 41 is the only prime for n=3.

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

%o (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

%Y Cf. A080076, A331539, A331540.

%K nonn

%O 1,2

%A _T. D. Noe_, Nov 17 2007

%E More terms from _Charles R Greathouse IV_, Mar 18 2010