Almost all positive integers have a property 
if the natural
density of the set satisfying 
is 1. If 
counts the positive integers
not exceeding 
that satisfy 
,
this means 
as 
. Equivalently, the number
of exceptions not exceeding 
is 
, where 
is one of the Landau symbols
of asymptotic notation.
For example, almost all positive integers are composite numbers (which is not in conflict with the second of Euclid's theorems that there are an infinite number of primes).
As another example, in any fixed integer base 
, almost all positive
integers contain any specified digit in their base-
representation. The decimal
digit 3 is an illustrative case (Numberphile 2012).
