Consider the recurrence relation
 |
(1)
|
with 
.
The first few iterations of 
for 
, 1, ... are 1, 1, 2, 3, 5, 10, 28, 154, 3520, ... (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).
The terms 
are integers up to 
but until 
which, rather surprisingly, is a rational number.
The numbers of decimal digits in the numerators of
for 
, 1, ... are 1, 1, 1, 1, 1, 2, 2, 3, 4, 7, 12, 22, 43, 85,
... (OEIS A389769). While an elegant approach
using congruences can show that 
must be rational (see below), the exact value 
was found explicitly by E. Weisstein (Oct. 13,
2025), following smaller initial computations by S. Wagon, in a calculation
requiring 370.9 gigabytes of memory. The exact value of 
is given by 
, where 
has 178485291570 decimal digits which require 74.1 gigabytes
to represent internally.
It is more convenient to work with the transformed sequence
 |
(4)
|
which gives the new recurrence
 |
(5)
|
with initial term 
.
The first few terms of this sequence for 
, 1, ... are 2, 6, 15, 40, 140, 924, 24640, ... (OEIS A061322). Now, 
will be nonintegral iff 
. The smallest prime
number 
for which 
(mod 
)
therefore gives the smallest nonintegral 
. In addition, since 
, 
will also be the smallest nonintegral 
.
For example, the first few sequences 
are summarized in the following table.
(Note that congruences applied to fractions yield
integer values.)
 |  |
| 2 | 0, 0 |
| 3 | 2, 0, 0 |
| 5 | 2, 1, 0, 0, 0 |
| 7 | 2, 6, 1, 5, 0, 0, 0 |
| 11 | 2, 6, 4, 7, 8, 0, 0, 0, 0, 0, 0 |
| 13 | 2, 6, 2, 1, 10, 1, 5, 1, 0, 0, 0, 0, 0 |
| 17 | 2, 6, 15, 6, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0 |
While 
(and therefore their values modulo 
) can be computed explicitly for small values of 
and 
, the fractions quickly grow too large to represent exactly.
However, computing terms modulo 
directly avoids term growth, and testing values of 
in this way shows that the first nonintegral 
is 
. As expected, direct verification of this fact is difficult
since
 |
(6)
|
(calculated using the asymptotic formula) is very large to be computed and stored explicitly. The first few values of 
for which 
are not integers are 43, 61, 67, 83, 103, 107, 109, 157,
... (OEIS A378851).
A sequence even more striking for assuming integer values only for finitely many terms is the 3-Göbel sequence
 |
(7)
|
The first few terms of this sequence are 1, 2, 5, 45, 22815, ... (OEIS A005166).
The Göbel sequences can be generalized to 
powers by
 |
(8)
|
Göbel's sequences can be generalized to 
-Göbel sequences, defined by the recursion
 |
(9)
|
for integers 
and with initial value 
(Ibstedt 1990, Gima et al. 2024).
-Göbel
Sequences." 14 Feb 2024.