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

7, 127, 2147483647, 170141183460469231731687303715884105727

Offset: 1

Author

Henry Bottomley, Nov 07 2002

Keywords

• nonn

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

Examples

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

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

Crossrefs

Cf. A077585 (double Mersenne numbers), A000225 (Mersenne numbers), A001348 (ditto with prime indices), A000040 (primes).

Programs

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 *)

PARI

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

Formula

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