An aspiring number is a positive integer that is not a perfect number but whose aliquot sequence eventually reaches a perfect number. Let
 |
where 
is the divisor function and 
is the restricted
divisor function, and define the aliquot sequence
of 
by
 |
If the sequence becomes constant at a positive value 
, then 
, so 
is a perfect number. For
example, beginning with 25 gives the sequence 25, 6,
6, 6, ..., so 25 is an aspiring number and 6 is a perfect
number.
The first few known aspiring numbers are 25, 95, 119, 143, ... (OEIS A063769). It is not known if 276 is an aspiring number, since the eventual behavior of its aliquot sequence remains unknown.
