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The multiplicity of a prime factor p of n is the largest exponent m for which p m divides n. The tables show the multiplicity for each prime factor. If no exponent is written then the multiplicity is 1 (since p = p 1). The multiplicity of a prime which does not divide n may be called 0 or may be considered undefined. Ω(n), the prime omega ...
For example, the prime factorization of the integer 60 is 60 = 2 × 2 × 3 × 5, the multiplicity of the prime factor 2 is 2, while the multiplicity of each of the prime factors 3 and 5 is 1. Thus, 60 has four prime factors allowing for multiplicities, but only three distinct prime factors.
In number theory, the prime omega functions and () count the number of prime factors of a natural number . Thereby (little omega) counts each distinct prime factor, whereas the related function () (big omega) counts the total number of prime factors of , honoring their multiplicity (see arithmetic function).
The prime number theorem is obtained there in an equivalent form that the Cesàro sum of the values of the Liouville function is zero. The Liouville function is () where () is the number of prime factors, with multiplicity, of the integer .
In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of the factors. [3] [4] [5] For example,
One of the simplest and most natural examples is the multiset of prime factors of a natural number n. Here the underlying set of elements is the set of prime factors of n . For example, the number 120 has the prime factorization 120 = 2 3 3 1 5 1 , {\displaystyle 120=2^{3}3^{1}5^{1},} which gives the multiset {2, 2, 2, 3, 5} .
However, it does not contain all the prime numbers, since the terms gcd(n + 1, a n) are always odd and so never equal to 2. 587 is the smallest prime (other than 2) not appearing in the first 10,000 outcomes that are different from 1. Nevertheless, in the same paper it was conjectured to contain all odd primes, even though it is rather inefficient.
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