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Ω(n), the prime omega function, is the number of prime factors of n counted with multiplicity (so it is the sum of all prime factor multiplicities). A prime number has Ω( n ) = 1. The first: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37 (sequence A000040 in the OEIS ).
For n ≥ 2, write the prime factorization of n in base 10 and concatenate the factors; ... 22 p − 1 ≡ 1 (mod p 2): 13, 673, 1595813, 492366587, 9809862296159 ...
Here the exponent () is the multiplicity of as a prime factor of (also known as the p-adic valuation of ). For example, in base 10, 378 = 2 1 · 3 3 · 7 1 is a Smith number since 3 + 7 + 8 = 2 · 1 + 3 · 3 + 7 · 1, and 22 = 2 1 · 11 1 is a Smith number, because 2 + 2 = 2 · 1 + (1 + 1) · 1.
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,
where is the th successive prime number, and all omitted terms (a 22 to a 228) are factors with exponent equal to one (i.e. the number is ). More concisely, it is the product of seven distinct primorials:
A Gaussian integer is either the zero, one of the four units (±1, ±i), a Gaussian prime or composite.The article is a table of Gaussian Integers x + iy followed either by an explicit factorization or followed by the label (p) if the integer is a Gaussian prime.
In number theory, the radical of a positive integer n is defined as the product of the distinct prime numbers dividing n. Each prime factor of n occurs exactly once as a factor of this product: r a d ( n ) = ∏ p ∣ n p prime p {\displaystyle \displaystyle \mathrm {rad} (n)=\prod _{\scriptstyle p\mid n \atop p{\text{ prime}}}p}
If one of the factors is composite, it can in turn be written as a product of smaller factors, for example 60 = 3 · 20 = 3 · (5 · 4). Continuing this process until every factor is prime is called prime factorization; the result is always unique up to the order of the factors by the prime factorization theorem.