%I #21 Apr 15 2022 15:46:57

%S 7,47,223,3967,16127,1046527,16769023,1073676287,68718952447,

%T 274876858367,4398042316799,1125899839733759,18014398241046527,

%U 1298074214633706835075030044377087

%N Primes of the form 4^n - 2^(n+1) - 1.

%C Cletus Emmanuel calls these "Carol primes".

%C There are only 25 such primes below 4^1000. Terms beyond a(15) are too large to be displayed here: The sequence should be extended by listing the corresponding n-values in A091515. - _M. F. Hasler_, May 15 2008

%C Is there an explanation for the following observed pattern? Between groups of primes of roughly the same size, there is a gap of about the magnitude of these primes, i.e., the size roughly doubles (e.g., after the 16- and 17-digit primes, there is a 34-digit prime, then a 78-digit prime and some others up to 105 digits, then some 200- to 250-digit primes, then approximately 500 digits...). - _M. F. Hasler_, May 15 2008

%H M. F. Hasler, <a href="/A091516/b091516.txt">Table of n, a(n) for n = 1..25</a>.

%H Ernest G. Hibbs, <a href="https://www.proquest.com/openview/4012f0286b785cd732c78eb0fc6fce80">Component Interactions of the Prime Numbers</a>, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Near-SquarePrime.html">Near-Square Prime</a>

%F a(k) = 4^A091515(k) - 2^(A091515(k) + 1) - 1 = (2^A091515(k) - 1)^2 - 2. - _M. F. Hasler_, May 15 2008

%t lst={};Do[p=(2^n-1)^2-2;If[PrimeQ[p], AppendTo[lst, p]], {n, 2, 160}];lst (* _Vladimir Joseph Stephan Orlovsky_, Sep 27 2008 *)

%o (PARI) c=0;for(n=1,999,ispseudoprime(4^n-2^(n+1)-1)&write("b091516.txt",c++," ",4^n-2^(n+1)-1)) \\ _M. F. Hasler_, May 15 2008

%Y Cf. A091515.

%K nonn

%O 1,1

%A _Eric W. Weisstein_, Jan 17 2004

%E Edited by _Ray Chandler_, Nov 15 2004