where
is a binomial coefficient. The first few
for , 1, 2, ... are 1, 5, 73, 1445, 33001,
819005, ... (OEIS A005259).
The first few prime Apéry numbers are 5, 73, 12073365010564729, 10258527782126040976126514552283001, ... (OEIS A092826), which have indices , 2, 12, 24, ... (OEIS A092825).
(Beukers 1987), where
is a generalized hypergeometric
function. The values for , 1, ... are 1, 3, 19, 147, 1251, 11253, 104959, ... (OEIS
A005258). The first few prime -numbers are 5, 73, 12073365010564729, 10258527782126040976126514552283001,
... (OEIS A092827), which have indices , 2, 6, 8, ... (OEIS A092828),
with no others for
(Weisstein, Mar. 8, 2004).
The numbers are also given by the recurrence
equation
(8)
with
and .
Both
and arose in Apéry's irrationality
proof of
and (van der Poorten 1979, Beukers
1987). They satisfy some surprising congruence properties,
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243-246, 1981.Beukers, F. "Some Congruences for the Apéry
Numbers." J. Number Th.21, 141-155, 1985.Beukers,
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V. "Binomial Sums and Identities." Maple Technical Newsletter10,
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der Poorten, A. "A Proof that Euler Missed... Apéry's Proof of the Irrationality
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