The factor problem

let us denote each number as the product of primes in a special way; for example
{2 3 5 7 11 ...}
{2 0 0 1 ...}
represents 28. (line up the primes in the first parentheses with the second one)
notice that in the prime factorization of 28, there a two 2's and one 7.
now, a further observation is that whenever we multiply two numbers, we add the number of each prime in that representation. for example 28 = [2 0 0 1] and 4 = [2 0 0 ...]. when we multiply them, we get 112 which is equal to [ 4 0 0 1]. similarly, when we raise a number to a power, we multiply each number in that "prime table" by that exponent. for example, 28^2 = [4 0 0 2], which is equal to 2x[2 0 0 1]. to summarize, multiplying numbers implies adding the corresponding prime tables and exponentiation of a number to a power implies multiplying every number in the prime table by the exponent.

now, the question is what addition of numbers translate to in the prime tables. all answers are appreciated.

#NumberTheory

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$