Schmidt (1993) proposed the problem of determining if for any integer 
, the sequence of numbers 
defined by the binomial
sums
 |
(1)
|
are all integers.
The following table gives the first few values of 
for small 
.
 | OEIS | values |
| 1 | A001850 | 1, 3, 13, 63, 321, 1683, 8989, 48639, ... |
| 2 | A005259 | 1, 5, 73, 1445, 33001, 819005, ... |
| 3 | A092813 | 1, 9, 433, 36729, 3824001, 450954009, ... |
| 4 | A092814 | 1, 17, 2593, 990737, 473940001, ... |
| 5 | A092815 | 1, 33, 15553, 27748833, 61371200001, ... |
This was proved by Strehl (1993, 1994) and Schmidt (1995) for the case 
, corresponding to the Franel
numbers. Strehl (1994) also found an explicit expression for the case 
. The resulting identities for 
are therefore known as the Strehl
identities. The problem was restated in Graham et al. (1994, pp. 256
and 549), who indicated that H. S. Wilf had shown 
to be an integer for any 
for 
(Zudilin 2004).
The problem was answered in the affirmative by Zudilin (2004), who found explicit expressions for all 
.
Particular cases include
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
with values for 
given by
 |
(7)
|
 |
(8)
|
(Zudilin 2004).
The following table summarizes the sequence of 
for small 
. Note that 
are precisely the Franel
numbers.
-Legendre Polynomials." J. Comput. Appl. Math. 49,
243-249, 1993.Schmidt, A. L. "Legendre Transforms and Apéry's
Sequences." J. Austral. Math. Soc. Ser. A 58, 358-375, 1995.Sloane,
N. J. A. Sequences