A077586 a(n) = 2^(2^prime(n) - 1) - 1.

7, 127, 2147483647, 170141183460469231731687303715884105727

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^p-1 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

Michael De Vlieger, Table of n, a(n) for n = 1..5

Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.

Luigi Morelli, DoubleMersennes.org

Eric Weisstein's World of Mathematics, Double Mersenne Number

Wikipedia, 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

Array[2^(2^Prime[#] - 1) - 1 &, 4] (* Michael De Vlieger, Apr 14 2022 *)

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