

A077586


a(n) = 2^(2^prime(n)  1)  1.


7




OFFSET

1,1


COMMENTS

First four terms are primes. Fifth (1.61585...*10^616), sixth (5.45374...*10^2465), seventh (2.007...*10^39456) and eighth (1.298...*10^157826) are not primes.
Note that a(n) divides 2^a(n)2 for every n, so if a(n) is composite then a(n) is a Fermat pseudoprime to base 2; cf. A007013.  Thomas Ordowski, Apr 08 2016
A number MM(p) is prime iff M(p) = A000225(p) = 2^p1 is a Mersenne prime exponent (A000043), which isn't possible unless p itself is also in A000043. Primes of this form are called double Mersenne primes MM(p). For all Mersenne exponents between 7 and 61, factors of MM(p) are known. The next candidate MM(61) is far too large to be merely stored on any existing hard drive (it would require 3*10^17 bytes), but a distributed search for factors of this and other MM(p) is ongoing, see the doublemersenne.org web site.  M. F. Hasler, Mar 05 2020


LINKS

Table of n, a(n) for n=1..4.
Eric Weisstein's World of Mathematics, Double Mersenne Number


FORMULA

a(n) = A077585(A000040(n)) = A000225(A001348(n)).


EXAMPLE

a(3) = 2^(2^5  1)  1 = 2^31  1 = 2147483647.


MAPLE

A077586 := n > 2^(2^ithprime(n)1)1; A077586(n) $ n=1..5; # M. F. Hasler, Mar 05 2020


MATHEMATICA

lst={}; Do[p=Prime[n]; If[PrimeQ[x=2^(2^p1)1], Print[x]; AppendTo[lst, n]], {n, 10^9}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 21 2008 *)


PROG

(PARI) apply( {A077586(n)=2^(2^prime(n)1)1}, [1..5]) \\ M. F. Hasler, Mar 05 2020


CROSSREFS

Cf. A077585 (double Mersenne numbers), A000225 (Mersenne numbers), A001348 (ditto with prime indices), A000040 (primes).
Sequence in context: A261487 A134722 A053713 * A277634 A309130 A263686
Adjacent sequences: A077583 A077584 A077585 * A077587 A077588 A077589


KEYWORD

nonn


AUTHOR

Henry Bottomley, Nov 07 2002


STATUS

approved



