%I #43 Apr 14 2022 11:19:50
%S 7,127,2147483647,170141183460469231731687303715884105727
%N a(n) = 2^(2^prime(n) - 1) - 1.
%C 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.
%C 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
%C 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
%H Michael De Vlieger, <a href="/A077586/b077586.txt">Table of n, a(n) for n = 1..5</a>
%H Ernest G. Hibbs, <a href="https://www.proquest.com/openview/4012f0286b785cd732c78eb0fc6fce80">Component Interactions of the Prime Numbers</a>, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
%H Luigi Morelli, <a href="http://www.doublemersennes.org/">DoubleMersennes.org</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DoubleMersenneNumber.html">Double Mersenne Number</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Double_Mersenne_number">Double Mersenne number</a>
%F a(n) = A077585(A000040(n)) = A000225(A001348(n)).
%e a(3) = 2^(2^5 - 1) - 1 = 2^31 - 1 = 2147483647.
%p A077586 := n -> 2^(2^ithprime(n)-1)-1; A077586(n) $ n=1..5; # _M. F. Hasler_, Mar 05 2020
%t Array[2^(2^Prime[#] - 1) - 1 &, 4] (* _Michael De Vlieger_, Apr 14 2022 *)
%o (PARI) apply( {A077586(n)=2^(2^prime(n)-1)-1}, [1..5]) \\ _M. F. Hasler_, Mar 05 2020
%Y Cf. A077585 (double Mersenne numbers), A000225 (Mersenne numbers), A001348 (ditto with prime indices), A000040 (primes).
%K nonn
%O 1,1
%A _Henry Bottomley_, Nov 07 2002