

A091516


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


9



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

1,1


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 nvalues 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 17digit primes, there is a 34digit prime, then a 78digit prime and some others up to 105 digits, then some 200 to 250digit 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, NearSquare Prime


FORMULA

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


MATHEMATICA

lst={}; Do[p=(2^n1)^22; If[PrimeQ[p], AppendTo[lst, p]], {n, 2, 160}]; lst (* Vladimir Joseph Stephan Orlovsky, Sep 27 2008 *)


PROG

(PARI) c=0; for(n=1, 999, ispseudoprime(4^n2^(n+1)1)&write("b091516.txt", c++, " ", 4^n2^(n+1)1)) \\ M. F. Hasler, May 15 2008


CROSSREFS

Cf. A091515.
Sequence in context: A202509 A009202 A093112 * A064385 A269520 A009260
Adjacent sequences: A091513 A091514 A091515 * A091517 A091518 A091519


KEYWORD

nonn


AUTHOR

Eric W. Weisstein, Jan 17 2004


EXTENSIONS

Edited by Ray Chandler, Nov 15 2004


STATUS

approved



