A proper cover is a cover of a set 
which does not have 
itself as a member (Macula 1994). Of the five covers
of 
, namely 
, 
, 
, 
, and 
, only 
does not have the set 
as a member and so is the unique
proper cover of two elements. For a labeled set of 
elements, where 
is a positive integer,
let 
and 
be the sets of all covers
and all proper covers, respectively. Then
 |  |  |
(1)
|
 |  | ![[1/2sum_(k=0)^(N)(-1)^k(N; k)2^(2^(N-k))]-(2^(2^N))/4,](/images/equations/ProperCover/Inline20.svg) |
(2)
|
giving 0, 1, 45, 15913, 1073579193, ... for 
, 2, ... (OEIS A007537).
