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m is a divisor of n (also called m divides n, or n is divisible by m) if all prime factors of m have at least the same multiplicity in n. The divisors of n are all products of some or all prime factors of n (including the empty product 1 of no prime factors). The number of divisors can be computed by increasing all multiplicities by 1 and then ...
The smallest abundant number not divisible by 2 or by 3 is 5391411025 whose distinct prime factors are 5, 7, 11, 13, 17, 19, 23, and 29 (sequence A047802 in the OEIS). An algorithm given by Iannucci in 2005 shows how to find the smallest abundant number not divisible by the first k primes. [1]
A positive divisor of that is different from is called a proper divisor or an aliquot part of (for example, the proper divisors of 6 are 1, 2, and 3). A number that does not evenly divide n {\displaystyle n} but leaves a remainder is sometimes called an aliquant part of n . {\displaystyle n.}
The Möbius function is defined by [3] = {= >The Möbius function can alternatively be represented as = () (),where is the Kronecker delta, () is the Liouville function, is the number of distinct prime divisors of , and () is the number of prime factors of , counted with multiplicity.
Note the set A does not have to be a set of prime factors, but it is typically a proper subset of the primes as seen in the factor base of Dixon's factorization method and the quadratic sieve. Likewise, it is what the general number field sieve uses to build its notion of smoothness, under the homomorphism ϕ : Z [ θ ] → Z / n Z ...
This means that the computation of greatest common divisor has, up to a constant factor, the same complexity as the multiplication. However, if a fast multiplication algorithm is used, one may modify the Euclidean algorithm for improving the complexity, but the computation of a greatest common divisor becomes slower than the multiplication.
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,
Equivalently, it is a number for which the sum of proper divisors (or aliquot sum) is less than n. For example, the proper divisors of 8 are 1, 2, and 4, and their sum is less than 8, so 8 is deficient. Denoting by σ(n) the sum of divisors, the value 2n – σ(n) is called the number's deficiency.