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

7, 47, 223, 3967, 16127, 1046527, 16769023, 1073676287, 68718952447, 274876858367, 4398042316799, 1125899839733759, 18014398241046527, 1298074214633706835075030044377087

Offset: 1

Author

Eric W. Weisstein, Jan 17 2004

Keywords

• nonn

Comments

Cletus Emmanuel calls these "Carol primes".

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

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

Links

• M. F. Hasler, Table of n, a(n) for n = 1..25.
• Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
• Eric Weisstein's World of Mathematics, Near-Square Prime

Crossrefs

Cf. A091515.

Programs

Mathematica

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 *)

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

Formula

a(k) = 4^A091515(k) - 2^(A091515(k) + 1) - 1 = (2^A091515(k) - 1)^2 - 2. - M. F. Hasler, May 15 2008

Extensions

Edited by Ray Chandler, Nov 15 2004