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A cyclic number [1] [2] is a natural number n such that n and φ(n) are coprime. Here φ is Euler's totient function. An equivalent definition is that a number n is cyclic if and only if any group of order n is cyclic. [3] Any prime number is clearly cyclic. All cyclic numbers are square-free. [4] Let n = p 1 p 2 …
A number n is called a cyclic number if Z/nZ is the only group of order n, which is true exactly when gcd(n, φ(n)) = 1. [13] The sequence of cyclic numbers include all primes, but some are composite such as 15. However, all cyclic numbers are odd except 2. The cyclic numbers are:
A cyclic compound (or ring compound) is a term for a compound in the field of chemistry in which one or more series of atoms in the compound is connected to form a ring.
A heterocyclic compound or ring structure is a cyclic compound that has atoms of at least two different elements as members of its ring(s). [1] Heterocyclic organic chemistry is the branch of organic chemistry dealing with the synthesis, properties, and applications of organic heterocycles. [2]
A simple ring contains the same number of sigma bonds as atoms, and a polycyclic ring system contains more sigma bonds than atoms. A molecule containing one or more rings is called a cyclic compound, and a molecule containing two or more rings (either in the same or different ring systems) is termed a polycyclic compound.
In between the prefix and the suffix, a pair of brackets with numerals denotes the number of carbon atoms between each of the bridgehead atoms. These numbers are arranged in descending order and are separated by periods. For example, the carbon frame of norbornane contains a total of 7 atoms, hence the root name heptane. This molecule has two ...
Cyclic numbers are related to the recurring digital representations of unit fractions. A cyclic number of length L is the digital representation of 1/(L + 1). Conversely, if the digital period of 1/p (where p is prime) is p − 1, then the digits represent a cyclic number. For example: 1/7 = 0.142857 142857...
In organic chemistry, the phenyl group, or phenyl ring, is a cyclic group of atoms with the formula C6H5, and is often represented by the symbol Ph (archaically φ) or Ø. The phenyl group is closely related to benzene and can be viewed as a benzene ring, minus a hydrogen, which may be replaced by some other element or compound to serve as a ...