Sylvester's sequence is the sequence defined by 
and the quadratic
recurrence equation
 |
(1)
|
for positive integer 
. This sequence is related to
Euclid's proof that there are an infinite number of
primes. Its recurrence
relation can also be written as
 |
(2)
|
for nonnegative integer 
(Vardi 1991). The terms are pairwise relatively
prime, so choosing one prime factor from each
term gives infinitely many distinct primes. There
is a constant
![E=1/2sqrt(6)exp{sum_(j=0)^infty2^(-j-2)ln[1+(2e_j-1)^(-2)]}=1.2640847353...](/images/equations/SylvestersSequence/NumberedEquation3.svg) |
(3)
|
(OEIS A076393) such that, for 
,
 |
(4)
|
where 
denotes the floor function (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 terms 
satisfy
 |
(5)
|
In addition, if 
is an irrational number, then each unit
fraction in its greedy algorithm representation
is at most 
,
where the first term is indexed by 
.
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 list
of factors less than 
of 
for 
and shows that 
is composite for 
. Furthermore, all numbers
less than 
in Sylvester's sequence are squarefree. Vardi (1991)
reported that no squareful numbers in this sequence
were known.
