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 2-almost prime because 17 and 113 are the only positive integers that can divide it without remainder other than 1 and 1921.

k-almost primality

A number can be said to be "k-almost prime" iff its prime factorization only has k non-unity terms (where k is some non-negative integer).

1-almost primes

Therefore, a prime number p whose only prime factorization is p = p * 1, then has k=1 (since there is only 1 non-unity factor) and can be said to be 1-almost prime which is by definition prime (despite the 'almost'). Therefore, under this defintion of "almost primality" prime numbers are simply the set of "1-almost prime" numbers.

n-almost primes

Whereas, a composite number like 4 = 2 * 2 * 1 has 2 non-unity prime factors and is hence a 2-almost prime.

The number 8 = 2 * 2 * 2 is a 3-almost prime, while 40 = 5 * 2 * 2 * 2 is a 4-almost prime.

1111111 = 4649 * 239 is a 2-almost prime

0-almost 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 unity-valued integers.

Since the prime factorization of 1 is 1 = 1 * 1 and of course this has 0 non-unity prime factors, then we are left with 1 (and all other unity factors) as the only 0-almost prime number.

Mersenne Almost-Primality

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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