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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
the k given prime numbers p i must be precisely the first k prime numbers (2, 3, 5, ...); if not, we could replace one of the given primes by a smaller prime, and thus obtain a smaller number than n with the same number of divisors (for instance 10 = 2 × 5 may be replaced with 6 = 2 × 3; both have four divisors);
The divisors of 10 illustrated with Cuisenaire rods: 1, 2, 5, and 10 In mathematics , a divisor of an integer n , {\displaystyle n,} also called a factor of n , {\displaystyle n,} is an integer m {\displaystyle m} that may be multiplied by some integer to produce n . {\displaystyle n.} [ 1 ] In this case, one also says that n {\displaystyle n ...
The above definition is unsuitable for defining gcd(0, 0), since there is no greatest integer n such that 0 × n = 0. However, zero is its own greatest divisor if greatest is understood in the context of the divisibility relation, so gcd(0, 0) is commonly defined as 0.
65536 is the natural number following 65535 and preceding 65537.. 65536 is a power of two: (2 to the 16th power).. 65536 is the smallest number with exactly 17 divisors (but there are smaller numbers with more than 17 divisors; e.g., 180 has 18 divisors) (sequence A005179 in the OEIS).
Number 1 is a unitary divisor of every natural number. The number of unitary divisors of a number n is 2 k, where k is the number of distinct prime factors of n. This is because each integer N > 1 is the product of positive powers p r p of distinct prime numbers p. Thus every unitary divisor of N is the product, over a given subset S of the ...
The abundancy index of n is the ratio σ(n)/n. [7] Distinct numbers n 1, n 2, ... (whether abundant or not) with the same abundancy index are called friendly numbers. The sequence (a k) of least numbers n such that σ(n) > kn, in which a 2 = 12 corresponds to the first abundant number, grows very quickly (sequence A134716 in the OEIS).
The abundancy index of n is the rational number σ(n) / n, in which σ denotes the sum of divisors function. A number n is a friendly number if there exists m ≠ n such that σ(m) / m = σ(n) / n. Abundancy is not the same as abundance, which is defined as σ(n) − 2n.