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A number that is not part of any friendly pair is called solitary. 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.
In mathematics, the amicable numbers are two different natural numbers related in such a way that the sum of the proper divisors of each is equal to the other number. That is, s(a)=b and s(b)=a, where s(n)=σ(n)-n is equal to the sum of positive divisors of n except n itself (see also divisor function). The smallest pair of amicable numbers is ...
In other words, (m, n) are a pair of betrothed numbers if s(m) = n + 1 and s(n) = m + 1, where s(n) is the aliquot sum of n: an equivalent condition is that σ(m) = σ(n) = m + n + 1, where σ denotes the sum-of-divisors function. The first few pairs of betrothed numbers (sequence A005276 in the OEIS) are: (48, 75), (140, 195), (1050, 1925 ...
The parity function maps a number to the number of 1's in its binary representation, modulo 2, so its value is zero for evil numbers and one for odious numbers. The Thue–Morse sequence , an infinite sequence of 0's and 1's, has a 0 in position i when i is evil, and a 1 in that position when i is odious.
In 1962, David Gale and Lloyd Shapley proved that, for any equal number of men and women, it is always possible to solve the stable marriage problem and make all marriages stable. They presented an algorithm to do so. [9] [10] The Gale–Shapley algorithm (also known as the deferred acceptance algorithm) involves a number of "rounds" (or ...
An even permutation can be obtained as the composition of an even number (and only an even number) of exchanges (called transpositions) of two elements, while an odd permutation can be obtained by (only) an odd number of transpositions. The following rules follow directly from the corresponding rules about addition of integers: [1]
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In mathematics, a pairing function is a process to uniquely encode two natural numbers into a single natural number. Any pairing function can be used in set theory to prove that integers and rational numbers have the same cardinality as natural numbers.