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_{k=1..2^n-1} binomial(2^n-1,k) = 2^(2^n-1) - 1 = A077585. Example: |W(n)| = a(n=2) = 7, because W = {[[1, 2]], [[1]], [[1, 2], [1]], [[1, 2], [2], [1]], [[1, 2], [2]], [[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
This is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for n, m >= 1. - Peter Bala, Dec 05 2022
LINKS
Eric Weisstein's World of Mathematics, Double Mersenne Number
FORMULA
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<<n-1)-1}, [0..9]) \\ M. F. Hasler, Mar 05 2020
(Python)
def A077585(n): return (1<<(1<<n)-1)-1 # Chai Wah Wu, Mar 14 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Henry Bottomley, Nov 07 2002
EXTENSIONS
Corrected by Lekraj Beedassy, Jan 02 2007
STATUS
approved