I have a better notation for prime factorisation (for some purposes). As follows:
and
then for any natural number n we have:
and now we can prove theorems involving prime factorisation by just using properties of d(k) and s(k).

I have a better notation for prime factorisation (for some purposes). As follows:
and
then for any natural number n we have:
and now we can prove theorems involving prime factorisation by just using properties of d(k) and s(k).
Do you realize that your "factorization" is, in general, not unique ? This is because you limited s to 0 or 1, making assumptions regarding P and d.Originally Posted by talanum1
In short this is worthless. It offers nothing beyond the usual expression of a natural number as a product of primes and actually offers less.
In what cases is it not unique?
I used s(k) as is so that a prime not occuring in the product can have a zero exponent.
The usual factorisation formula would have the same strength of notation only if you correlate the sub and superscript sequences, so my notation is about the same worth in that case (if you take s(k) as mapping to ).
One of the basic concepts of mathematics is expressing numbers in terms of smaller numbers, whether by addition (seven is the sum of three and four), multiplication (12 is the product of three and four), or exponentiation (81 is three to the fourth power). We often find it useful to break a number down into smaller numbers. Given a number, we call any number that divides evenly into it one of its factors, where dividing evenly means there is no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6 and 12, because 12 can be divided by each of these with no remainder.
It is often useful to write a number in terms of its prime factorization, or as the product of its prime factors. For example, 56 can be written as 2×2×2×7 and 84 can be written as 2×2×3×7 . Every number can be written as a product of primes, and, like a fingerprint, every number has a unique prime factorization.
Example . Compute the prime factorization of 23,100.
Step 1. 23, 100/2 = 11, 550 . Write down 2.
Step 2. 11, 550/2 = 5, 775 . Write down 2.
Step 3. 5, 775/3 = 1, 925 . Write down 3.
Step 4. 1, 925/5 = 385 . Write down 5.
Step 5. 385/5 = 77 . Write down 5.
Step 6. 77/7 = 11 . Write down 7.
Step 7. 11 is prime. Write down 11.
Therefore, the prime factorization of 23,100 is 2×2×3×5×5×7×11 .
So now i think now it is clear what is a prime factorization
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