Consider the recurrence relation
(1)
|
with . The first few iterates of are 1, 2, 3, 5, 10, 28, 154, ... (OEIS A003504). The terms grow extremely rapidly, but are given by the asymptotic formula
(2)
|
(OEIS A116603; correcting Finch 2003, p. 446), where
(3)
|
(OEIS A115632; Finch 2003, p. 446; Zagier). It is more convenient to work with the transformed sequence
(4)
|
which gives the new recurrence
(5)
|
with initial condition . Now, will be nonintegral iff . The smallest for which (mod ) therefore gives the smallest nonintegral . In addition, since , is also the smallest nonintegral .
For example, we have the sequences :
(6)
|
(7)
|
(8)
|
Testing values of shows that the first nonintegral is . Note that a direct verification of this fact is impossible since
(9)
|
(calculated using the asymptotic formula) is much too large to be computed and stored explicitly.
A sequence even more striking for assuming integer values only for finitely many terms is the 3-Göbel sequence
(10)
|
The first few terms of this sequence are 1, 2, 5, 45, 22815, ... (OEIS A005166).
The Göbel sequences can be generalized to powers by
(11)
|