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d() is the number of positive divisors of n, including 1 and n itself; σ() is the sum of the positive divisors of n, including 1 and n itselfs() is the sum of the proper divisors of n, including 1 but not n itself; that is, s(n) = σ(n) − n
For example, to find the Hall divisors of 60, its prime power factorization is 2 2 × 3 × 5, so one takes any product of 3, 2 2 = 4, and 5. Thus, the Hall divisors of 60 are 1, 3, 4, 5, 12, 15, 20, and 60. A Hall subgroup of G is a subgroup whose order is a Hall divisor of the order of G. In other words, it is a subgroup whose order is coprime ...
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.
In number theory, the aliquot sum s(n) of a positive integer n is the sum of all proper divisors of n, that is, all divisors of n other than n itself. That is, = |,. It can be used to characterize the prime numbers, perfect numbers, sociable numbers, deficient numbers, abundant numbers, and untouchable numbers, and to define the aliquot sequence of a number.
Since the sum of its divisors (excluding the number itself) 2040 > 840; It is an abundant number and also a superabundant number. [2] It is an idoneal number. [3] It is the least common multiple of the numbers from 1 to 8. [4] It is the smallest number divisible by every natural number from 1 to 10, except 9.
For every divisor d of n, G has at most one subgroup of order d. If either (and thus both) are true, it follows that there exists exactly one subgroup of order d, for any divisor of n. This statement is known by various names such as characterization by subgroups. [5] [6] [7] (See also cyclic group for some characterization.)
The oldforms for Γ 0 (N) are those modular forms f(τ) of level N of the form g(d τ) for modular forms g of level M with M a proper divisor of N, where d divides N/M. The newforms are defined as a vector subspace of the modular forms of level N , complementary to the space spanned by the oldforms, i.e. the orthogonal space with respect to the ...
The greatest common divisor (GCD) of integers a and b, at least one of which is nonzero, is the greatest positive integer d such that d is a divisor of both a and b; that is, there are integers e and f such that a = de and b = df, and d is the largest such integer.