Foias constant

In mathematical analysis, the Foias constant, is a number named after Ciprian Foias.

If x₁ > 0 and

then the Foias constant is the unique real number α such that if x₁ = α then the sequence diverges to ∞. Numerically, it is

No closed form is known.

When x₁ = α then we have the limit:

where "log" denotes the usual natural logarithm.

A fortuitous observation between the prime number theorem and this constant goes as follows,

where π is the prime-counting function.