The Foias constant is the unique positive real number 
for which the recurrence relation
 |
(1)
|
for 
,
2, ..., with 
,
generates a sequence satisfying 
(Ewing and Foias 2000).
The question of whether a positive initial value can make this sequence tend to infinity appeared in a fall issue of Gazeta Matematică in the mid-1970s. It was listed as a problem from an entrance exam for prospective freshmen in the mathematics department at the University of Bucharest and was solved by C. Foias (Ewing and Foias 2000).
The constant has approximate value
 |
(2)
|
(OEIS A085848). No closed form is known for this constant. For the initial value 
,
 |
(3)
|
which can be rewritten as
 |
(4)
|
where 
is the prime counting function. However,
Ewing and Foias (2000) believe that this connection with the prime
number theorem is fortuitous.
Foias also discovered that the problem stated in the journal was a misprint of the actual exam problem, which used the recurrence
relation 
(Ewing and Foias 2000). In this form, the recurrence converges to
 |
(5)
|
(OEIS A085846), which is simply the root of
 |
(6)
|
for all positive starting values 
.
