cp's OEIS Frontend

A091516 Primes of the form 4^n - 2^(n+1) - 1.

%I A091516 #22 Feb 16 2025 08:32:52
%S A091516 7,47,223,3967,16127,1046527,16769023,1073676287,68718952447,
%T A091516 274876858367,4398042316799,1125899839733759,18014398241046527,
%U A091516 1298074214633706835075030044377087
%N A091516 Primes of the form 4^n - 2^(n+1) - 1.
%C A091516 Cletus Emmanuel calls these "Carol primes".
%C A091516 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 A091516 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 A091516 M. F. Hasler, <a href="/A091516/b091516.txt">Table of n, a(n) for n = 1..25</a>.
%H A091516 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 A091516 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Near-SquarePrime.html">Near-Square Prime</a>
%F A091516 a(k) = 4^A091515(k) - 2^(A091515(k) + 1) - 1 = (2^A091515(k) - 1)^2 - 2. - _M. F. Hasler_, May 15 2008
%t A091516 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 A091516 (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 A091516 Cf. A091515.
%K A091516 nonn
%O A091516 1,1
%A A091516 _Eric W. Weisstein_, Jan 17 2004
%E A091516 Edited by _Ray Chandler_, Nov 15 2004