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The notations d(n), ν(n) and τ(n) (for the German Teiler = divisors) are also used to denote σ 0 (n), or the number-of-divisors function [1] [2] (OEIS: A000005). When z is 1, the function is called the sigma function or sum-of-divisors function , [ 1 ] [ 3 ] and the subscript is often omitted, so σ ( n ) is the same as σ 1 ( n ) ( OEIS ...
In number theory, a weird number is a natural number that is abundant but not semiperfect. [ 1 ] [ 2 ] In other words, the sum of the proper divisors ( divisors including 1 but not itself) of the number is greater than the number, but no subset of those divisors sums to the number itself.
Prime numbers have exactly 2 divisors, and highly composite numbers are in bold. 7 is a divisor of 42 because =, so we can say It can also be said that 42 is divisible by 7, 42 is a multiple of 7, 7 divides 42, or 7 is a factor of 42. The non-trivial divisors of 6 are 2, −2, 3, −3.
The easiest way to find the Hall divisors is to write the prime power factorization of the number in question and take any subset of the factors. 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 ...
Greatest common divisors can be computed by determining the prime factorizations of the two numbers and comparing factors. For example, to compute gcd(48, 180), we find the prime factorizations 48 = 2 4 · 3 1 and 180 = 2 2 · 3 2 · 5 1; the GCD is then 2 min(4,2) · 3 min(1,2) · 5 min(0,1) = 2 2 · 3 1 · 5 0 = 12 The corresponding LCM is ...
a composite number has more than just 1 and itself as divisors; that is, d(n) > 2; a highly composite number has a number of positive divisors that is greater than any lesser number; that is, d(n) > d(m) for every positive integer m < n. Counterintuitively, the first two highly composite numbers are not composite numbers.
The smallest friendly number is 6, forming for example, the friendly pair 6 and 28 with abundancy σ(6) / 6 = (1+2+3+6) / 6 = 2, the same as σ(28) / 28 = (1+2+4+7+14+28) / 28 = 2. The shared value 2 is an integer in this case but not in many other cases. Numbers with abundancy 2 are also known as perfect numbers. There are several unsolved ...
There is a larger class of number-theoretic functions that do not fit this definition, for example, the prime-counting functions. This article provides links to functions of both classes. An example of an arithmetic function is the divisor function whose value at a positive integer n is equal to the number of divisors of n.