Göbel's sequence is defined by the recurrence relation
 |
(1)
|
for 
,
with 
.
This indexing gives 
for 
in the notation of Gima et al. (2024). The first few terms 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 
is a nonintegral rational
number (Gima et al. 2024). 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 
is not an integer (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 relation
 |
(5)
|
for 
,
with initial term 
.
The first few terms for 
, 2, ... are 2, 6, 15, 40, 140, 924, 24640, ... (OEIS A061322). If 
is an integer, then 
is nonintegral iff 
. For a prime number 
, a nonzero residue of 
modulo 
shows that 
is nonintegral. Checking only prime
numbers does not by itself rule out an earlier failure at a composite index;
prime power divisibility must also be checked (Zagier).
For example, the first few sequences 
are summarized in the following table.
Residues of rational numbers whose denominators
are not divisible by 
are computed using the multiplicative inverse
of the denominator modulo 
.
 |  |
| 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 the residue for 
proves that 
is nonintegral. 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 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)
|
for 
,
with 
.
The first few terms are 1, 2, 5, 45, 22815, ... (OEIS A005166).
The Göbel sequences can be generalized to integer powers 
by
 |
(8)
|
for 
,
with 
.
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.