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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.
where n > 1 is an integer and p, q, r are prime numbers, then 2 n × p × q and 2 n × r are a pair of amicable numbers. This formula gives the pairs (220, 284) for n = 2, (17296, 18416) for n = 4, and (9363584, 9437056) for n = 7, but no other such pairs are known. Numbers of the form 3 × 2 n − 1 are known as Thabit numbers.
N s, the number of pairs of cases ranked in the same order on both variables (number of concordant pairs), N d, the number of pairs of cases ranked in reversed order on both variables (number of reversed pairs), where "ties" (cases where either of the two variables in the pair are equal) are dropped. Then
All known pairs of betrothed numbers have opposite parity. Any pair of the same parity must exceed 10 10. ... Handbook of Number Theory I. Dordrecht: ...
The number of discordant pairs equals N D = 1 × 5 + 1 × 2 + 7 × 2 = 21. {\displaystyle N_{D}=1\times 5+1\times 2+7\times 2=21.} The number of pairs tied is equal to the total number of pairs minus the concordant and discordant pairs
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This is equal to the number of electrons in its valence shell if all the valence shell electrons are used for bonding. If they are not, the remainder will form non-bonding electron pairs, usually known as lone pairs. The valence of a bond, S, is defined as the number of electron pairs forming the bond. In general this is not an integral number.
Indeed, this same technique can also be followed to try and derive any number of other functions for any variety of schemes for enumerating the plane. A pairing function can usually be defined inductively – that is, given the n th pair, what is the (n+1) th pair? The way Cantor's function progresses diagonally across the plane can be expressed as