A prime number has no integer factors between itself and 1, meaning the only way to represent it as a multiple of integers is to represent it as 1 * N  where N is the number itself.
Meanwhile, a number like 1921 is not prime, because 17*113 = 1921, but 1921 is a 2almost prime because 17 and 113 are the only positive integers that can divide it without remainder other than 1 and 1921.
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kalmost primality
A number can be said to be "kalmost prime" iff its prime factorization only has k nonunity terms (where k is some nonnegative integer).
1almost primes
Therefore, a prime number p whose only prime factorization is p = p * 1, then has k=1 (since there is only 1 nonunity factor) and can be said to be 1almost prime which is by definition prime (despite the 'almost'). Therefore, under this defintion of "almost primality" prime numbers are simply the set of "1almost prime" numbers.
nalmost primes
Whereas, a composite number like 4 = 2 * 2 * 1 has 2 nonunity prime factors and is hence a 2almost prime.
The number 8 = 2 * 2 * 2 is a 3almost prime, while 40 = 5 * 2 * 2 * 2 is a 4almost prime.
1111111 = 4649 * 239 is a 2almost prime
0almost primes
One benefit of this definition of 'almost primality' is that it allows prime numbers and composite numbers to be separated into separate groups, while also ensuring that both groups are distinct from the ambiguous nature of unityvalued integers.
Since the prime factorization of 1 is 1 = 1 * 1 and of course this has 0 nonunity prime factors, then we are left with 1 (and all other unity factors) as the only 0almost prime number.
Mersenne AlmostPrimality
(see also Category:Mersenne Primes#Almost Mersenne Primes)
A Mersenne prime is a number of the form:

 for
 that are also prime
e.g.
Almost Mersenne Primes
An 'almost Mersenne prime' (aMp) can be defined as a number of the form:
 that is not a prime number ^{(equiv.: 'is an almost prime number')}
e.g.
Each 'Almost Mersenne Prime' has a corresponding Almost Perfect Number (in the same way each Mersenne Prime has a corresonding Perfect Number).
PID = 4815 [= 18 [= Lp9
 4815 = 107 * 5 * 3^{^2}
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