An Apéry number prime is a prime number occurring in either of the Apéry number sequences 
and 
. These sequences are defined for 
by
 |
and
 |
where 
is a binomial coefficient.
The first few prime 
values are 5, 73, 12073365010564729, 10258527782126040976126514552283001,
... (OEIS A092826), occurring at indices 
, 2, 12, 24, .... The next listed index
is 6624 (OEIS A092825). The corresponding 
has 10136 digits
and is a probable prime.
The first few prime 
values are 3, 19, 104959, 9793891, ... (OEIS A092828),
occurring at indices 
,
2, 6, 8, .... In particular, 
is a proved prime number.
The Apéry number sequences are distinct from decimal prefixes of Apéry's constant. Apéry-constant primes are formed from those prefixes.
