Sylvester's Sequence
The sequence defined by 
and the quadratic recurrence equation
 |
(1)
|
This sequence arises in Euclid's proof that there are an infinite number of primes. The proof proceeds by constructing a sequence of primes using the recurrence relation
 |
(2)
|
(Vardi 1991). Amazingly, there is a constant
![E=1/2sqrt(6)exp{sum_(j=1)^infty2^(-j-1)ln[1+(2e_j-1)^(-2)]}=1.2640847353...](/images/equations/SylvestersSequence/NumberedEquation3.gif) |
(3)
|
(OEIS A076393) such that
 |
(4)
|
(Aho and Sloane 1973, Vardi 1991, Graham et al. 1994). The first few numbers in Sylvester's sequence are 2, 3, 7, 43, 1807, 3263443, 10650056950807, ... (OEIS
A000058). The 
satisfy
 |
(5)
|
In addition, if 
is an irrational
number, then the 
th term of an infinite
sum of unit fractions used to represent 
as computed using
the greedy algorithm must be smaller than 
.
The 
of the first few prime 
are 0, 1, 2, 3, 5, ..., corresponding to 2, 3,
7, 43, 3263443, ... (OEIS A014546). Vardi (1991)
gives a lists of factors less than 
of 
for 
and shows
that 
is composite
for 
. Furthermore, all numbers
less than 
in Sylvester's sequence
are squarefree, and no squareful
numbers in this sequence are known (Vardi 1991).
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