query: subclass_of sphenic number
supports: Number theory or, in older usage, arithmetic is a branch of pure mathematics devoted primarily to the study of the integers. It is sometimes called "The Queen of Mathematics" because of its foundational place in the discipline. Number theorists study prime numbers as well as the properties of objects made out of integers (e.g., rational numbers) or defined as generalizations of the integers (e.g., algebraic integers). An integer (from the Latin "integer" meaning "whole") is a number that can be written without a fractional component. For example, 21, 4, 0, and 2048 are integers, while 9.75, , and  are not. In Algebraic number theory, an algebraic integer is a complex number that is a root of some monic polynomial (a polynomial whose leading coefficient is 1) with coefficients in formula_1 (the set of integers). The set of all algebraic integers, , is closed under addition and multiplication and therefore is a commutative subring of the complex numbers. The ring is the integral closure of regular integers formula_1 in complex numbers. A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example, 5 is prime because 1 and 5 are its only positive integer factors, whereas 6 is composite because it has the divisors 2 and 3 in addition to 1 and 6. The fundamental theorem of arithmetic establishes the central role of primes in number theory: any integer greater than 1 can be expressed as a product of primes that is unique up to ordering. The uniqueness in this theorem requires excluding 1 as a prime because one can include arbitrarily many instances of 1 in any factorization, e.g., 3, 1 · 3, 1 · 1 · 3, etc. are all valid factorizations of 3. In number theory , a sphenic number ( from Ancient Greek :  , ' wedge ' ) is a positive integer that is the product of three distinct prime numbers . The smallest sphenic number is 30 = 2 × 3 × 5 , the product of the smallest three primes . Note that this definition is more stringent than simply requiring the integer to have exactly three prime factors ; e.g. 60 = 22 × 3 × 5 has exactly 3 prime factors , but is not sphenic . All sphenic numbers have exactly eight divisors . If we express the sphenic number as n = p \ cdot q \ cdot r , where p , q , and r are distinct primes , then the set of divisors of n will be : \ left \ ( 1 , \ p , \ q , \ r , \ pq , \ pr , \ qr , \ n \ right \ ) . All sphenic numbers are by definition squarefree , because the prime factors must be distinct . The Möbius function of any sphenic number is  1 . The cyclotomic polynomials \ Phi_n ( x ) , taken over all sphenic numbers n , may contain arbitrarily large coefficients ( for n a product of two primes the coefficients are \ pm 1 or 0 ) . The first few sphenic numbers are : 30 , 42 , 66 , 70 , 78 , 102 , 105 , 110 , 114 , 130 , 138 , 154 , 165 , ... ( sequence A007304 in OEIS ) As of January 2016 the largest known sphenic number is ( 274,207,281  1 ) × ( 257,885,161  1 ) × ( 243,112,609  1 ) , i.e. , the product of the three largest known primes . In mathematics, a rational number is any number that can be expressed as the quotient or fraction "p"/"q" of two integers, a numerator "p" and a non-zero denominator "q". Since "q" may be equal to 1, every integer is a rational number. The set of all rational numbers, often referred to as "the rationals", is usually denoted by a boldface Q (or blackboard bold formula_1, Unicode ); it was thus denoted in 1895 by Giuseppe Peano after "quoziente", Italian for "quotient". In number theory, the prime factors of a positive integer are the prime numbers that divide that integer exactly. The prime factorization of a positive integer is a list of the integer's prime factors, together with their multiplicities; the process of determining these factors is called integer factorization. The fundamental theorem of arithmetic says that every positive integer has a single unique prime factorization. Broadly speaking, pure mathematics is mathematics that studies entirely abstract concepts. This was a recognizable category of mathematical activity from the 19th century onwards, at variance with the trend towards meeting the needs of navigation, astronomy, physics, economics, engineering, and so on.
candidates: algebraic number branch category complex number engineering fraction integer latin mathematics meeting natural natural number number out polynomial product rational number set study theory
answer: natural number