Prime signature
Encyclopedia
The prime signature of a number is the multiset
of exponents of its prime factorisation.
For example, all prime number
s have a prime signature of {1}, the squares of primes have a prime signature of {2}, the products of 2 distinct primes have a prime signature of {1,1} and the products of a square of a prime and a different prime (e.g. 12,18,20,... ) have a prime signature of {2,1}.
The number of divisor
s that a number has is determined by its prime signature as follows: If you add one to each exponent and multiply them together you get the number of divisors including the number itself and 1. For example, 20 has prime signature {2,1} and so the number of divisors is 3x2=6. They are 1,2,4,5,10 and 20.
The smallest number of each prime signature is a product of primorial
s. The first few are:
Multiset
In mathematics, the notion of multiset is a generalization of the notion of set in which members are allowed to appear more than once...
of exponents of its prime factorisation.
For example, all 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 have a prime signature of {1}, the squares of primes have a prime signature of {2}, the products of 2 distinct primes have a prime signature of {1,1} and the products of a square of a prime and a different prime (e.g. 12,18,20,... ) have a prime signature of {2,1}.
The number of divisor
Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...
s that a number has is determined by its prime signature as follows: If you add one to each exponent and multiply them together you get the number of divisors including the number itself and 1. For example, 20 has prime signature {2,1} and so the number of divisors is 3x2=6. They are 1,2,4,5,10 and 20.
The smallest number of each prime signature is a product of primorial
Primorial
In mathematics, and more particularly in number theory, primorial is a function from natural numbers to natural numbers similar to the factorial function, but rather than multiplying successive positive integers, only successive prime numbers are multiplied...
s. The first few are:
- 1, 2, 4, 6, 8, 12, 16, 24, 30, 32, 36, 48, 60, 64, 72, 96, 120, 128, 144, 180, 192, 210, 216, ... .
Numbers with same prime signature
| Signature | Numbers | OEIS On-Line Encyclopedia of Integer Sequences The On-Line Encyclopedia of Integer Sequences , also cited simply as Sloane's, is an online database of integer sequences, created and maintained by N. J. A. Sloane, a researcher at AT&T Labs... ID | Description |
|---|---|---|---|
| {1} | 2, 3, 5, 7, 11, ... | prime numbers | |
| {2} | 4, 9, 25, 49, 121, ... | squares Square number In mathematics, a square number, sometimes also called a perfect square, is an integer that is the square of an integer; in other words, it is the product of some integer with itself... of prime numbers |
|
| {1,1} | 6, 10, 14, 15, 21, ... | two distinct prime divisors (square-free 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... semiprime Semiprime In mathematics, a semiprime is a natural number that is the product of two prime numbers. The first few semiprimes are 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, ... .... s) |
|
| {3} | 8, 27, 125, 343, ... | cubes of prime numbers | |
| {2,1} | 12, 18, 20, 28, ... | squares of primes times another prime | |
| {4} | 16, 81, 625, 2401, ... | fourth powers of prime numbers | |
| {3,1} | 24, 40, 54, 56, ... | cubes of primes times another prime | |
| {1,1,1} | 30, 42, 66, 70, ... | three distinct prime divisors (sphenic number Sphenic number In number theory, a sphenic number is a positive integer which is the product of three distinct prime numbers.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... s) |
|
| {5} | 32, 243, 3125, ... | fifth powers of primes | |
| {2,2} | 36, 100, 196, 225, ... | squares of square-free semiprimes |
Sequences defined by their prime signature
Given a number with prime signature S, it is- A prime numberPrime numberA 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...
if S = {1} - A square if gcd S is even
- A square-free integerSquare-free integerIn 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...
if max S = 1 - A powerful numberPowerful numberA powerful number is a positive integer m such that for every prime number p dividing m, p2 also divides m. Equivalently, a powerful number is the product of a square and a cube, that is, a number m of the form m = a2b3, where a and b are positive integers. Powerful numbers are also known as...
if min S ≥ 2 - An Achilles numberAchilles numberAn Achilles number is a number that is powerful but not a perfect power. A positive integer n is a powerful number if, for every prime divisor or factor p of n, p2 is also a divisor. In other words, every prime factor appears at least squared. All Achilles numbers are powerful...
if min S ≥ 2 and gcd S = 1 - k-almost prime if sum S = k.