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 A077585 a(n) = 2^(2^n-1) - 1. 12
 0, 1, 7, 127, 32767, 2147483647, 9223372036854775807, 170141183460469231731687303715884105727, 57896044618658097711785492504343953926634992332820282019728792003956564819967 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS a(2), a(3), a(5) and a(7) are prime; a(11) is not. Let S be a set of n elements. First we perform a set partition on S. Let SU be the set of all nonempty subsets of S. As is well known, 2^n - 1 is the number of nonempty subsets of a set with n elements (see A000225). That is, 2^n - 1 is the number of elements |SU| of SU. In the second step, we select k elements from SU. We want to know how many different selections are possible. Let W be the resulting set of selections formed from SU. Then the number of elements |W| of W is |W| = sum((binomial(2^n-1,k)), k=1..2^n-1) = 2^(2^n-1)-1 = A077585. Example: |W(n)| = a(n=2) = 7, because W={[[1, 2]], [], [[1, 2], ], [[1, 2], , ], [[1, 2], ], [, ], []}. - Thomas Wieder, Nov 08 2007 These are so-called double Mersenne numbers, sometimes denoted MM(n) where M = A000225. MM(n) is prime iff M(n) = 2^n-1 is a Mersenne prime exponent (A000043), which isn't possible unless n 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. MM(61) is far too large for any currently known primality test, 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 Eric Weisstein's World of Mathematics, Double Mersenne Number FORMULA a(n) = A000225(A000225(n)). a(n) = A058891(n+1)-1. - corrected by Maurizio De Leo, Feb 25 2015 a(n) = (A001146(n)-2)/2. a(n) = A056220(1+a(n-1)). a(n) = Sum_{k=1..2^n-1} binomial(2^n-1,k). - Thomas Wieder, Nov 08 2007 a(n) = 2 * a(n-1)^2 + 4 * a(n-1) + 1. - Roderick MacPhee, Oct 05 2012 EXAMPLE a(5) = 2^(2^5-1)-1 = 2^31-1 = 2147483647. MAPLE a:= n-> 2^(2^n-1)-1: seq(a(n), n=0..8); # Thomas Wieder, Nov 08 2007 MATHEMATICA 2^(2^Range[0, 9] - 1) - 1 (* Vladimir Joseph Stephan Orlovsky, Jun 22 2011 *) PROG (PARI) a(n)=if(n<1, 0, -1+2*(1+a(n-1))^2) (PARI) apply( {A077585(n)=1<<(1<

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Last modified September 18 07:39 EDT 2020. Contains 337166 sequences. (Running on oeis4.)