Sphenic number

Sphenic number

Discussion

Encyclopedia
In number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

, a sphenic number (from  — wedge) is a positive integer
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...

which is the product of three distinct prime number
Prime number
A prime number 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, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

s.

Note that this definition is more stringent than simply requiring the integer to have exactly three prime factor
Prime factor
In number theory, the prime factors of a positive integer are the prime numbers that divide that integer exactly, without leaving a remainder. The process of finding these numbers is called integer factorization, or prime factorization. A prime factor can be visualized by understanding Euclid's...

s; 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 , where p, q, and r are distinct primes, then the set of divisors of n will be:

All sphenic numbers are by definition squarefree
Square-free integer
In mathematics, a square-free, or quadratfrei, integer is one divisible by no perfect square, except 1. For example, 10 is square-free but 18 is not, as it is divisible by 9 = 32...

, because the prime factors must be distinct.

The Möbius function
Möbius function
The classical Möbius function μ is an important multiplicative function in number theory and combinatorics. The German mathematician August Ferdinand Möbius introduced it in 1832...

of any sphenic number is −1.

The cyclotomic polynomials , taken over all sphenic numbers n, may contain arbitrarily large coefficients (for n a product of two primes the coefficients are or 0).

The first few sphenic numbers are: 30
30 (number)
30 is the natural number following 29 and preceding 31.-In mathematics:30 is the sum of the first four squares, which makes it a square pyramidal number.It is a primorial and is the smallest Giuga number....

, 42
42 (number)
42 is the natural number immediately following 41 and directly preceding 43. The number has received considerable attention in popular culture as a result of its central appearance in The Hitchhiker's Guide to the Galaxy as the "Answer to the Ultimate Question of Life, the Universe, and...

, 66
66 (number)
66 is the natural number following 65 and preceding 67.Usages of this number include:-Mathematics:*66 is a sphenic number, a triangular number, a hexagonal number, and a semi-meandric number...

, 70
70 (number)
70 is the natural number following 69 and preceding 71.-In mathematics:Its factorization makes it a sphenic number. 70 is a Pell number and a generalized heptagonal number, one of only two numbers to be both. Also, it is the seventh pentagonal number and the fourth 13-gonal number, as well as the...

, 78
78 (number)
78 is the natural number following 77 and followed by 79.-In mathematics:78 is a triangular number, and its factorization makes it a sphenic number...

, 102
102 (number)
102 is the natural number following 101 and preceding 103.-In mathematics:102 is an abundant number and semiperfect number. It is a sphenic number...

, 105
105 (number)
105 is the natural number following 104 and preceding 106.-In mathematics:105 is a triangular number, a 12-gonal number and a Zeisel number. It is a sphenic number, and is the product of three consecutive prime numbers. 105 is the double factorial of 7...

, 110
110 (number)
110 is the natural number following 109 and preceding 111.It is also known as "eleventy", a term made famous by linguist and author J. R. R...

, 114
114 (number)
114 is the natural number following 113 and preceding 115.-In mathematics:*One hundred [and] fourteen is an abundant number, a sphenic number and a Harshad number. It is the sum of the first four hyperfactorials, including H...

, 130
130 (number)
130 is the natural number following 129 and preceding 131.-In mathematics:130 is a sphenic number. It is a noncototient since there is no answer to the equation x - φ = 130....

, 138
138 (number)
138 is the natural number following 137 but before 139.-In mathematics:* Its factorization makes 138 a sphenic number* The sum of four consecutive primes...

, 154
154 (number)
One hundred and fifty-four is the natural number following one hundred and fifty-three and preceding one hundred and fifty-five.-In mathematics:* 154 is a nonagonal number...

, 165
165 (number)
165 is the natural number following 164 and preceding 166.-In mathematics:* 165 is an odd number* 165 is a composite number* 165 is a deficient number* 165 is a binary palindromic number * 165 is a sphenic number...

, ...

The first case of two consecutive integers which are sphenic numbers is 230 = 2×5×23 and 231 = 3×7×11. The first case of three is 1309 = 7×11×17, 1310 = 2×5×131, and 1311 = 3×19×23. There is no case of more than three, because every fourth consecutive positive integer is divisible by 4 = 2×2 and therefore not squarefree.

the largest known sphenic number is (243,112,609 − 1) × (242,643,801 − 1) × (237,156,667 − 1), i.e., the product of the three largest known prime
Largest known prime
The largest known prime number is the largest integer that is currently known to be a prime number.It was proven by Euclid that there are infinitely many prime numbers; thus, there is always a prime greater than the largest known prime. Many mathematicians and hobbyists search for large prime numbers...

s.