mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the prime signature of a number is the

multiset In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that e ...

of (nonzero) exponents of its

prime factorization In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization. When the numbers are suf ...

. The prime signature of a number having prime factorization

p_1^p_2^ \dots p_n^

is the multiset

\left \

. For example, all

prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

s have a prime signature of , the

squares In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...

of primes have a prime signature of , the products of 2 distinct primes have a prime signature of and the products of a square of a prime and a different prime (e.g. 12, 18, 20, ...) have a prime signature of .

Properties

The

divisor function In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as ''the'' divisor function, it counts the ''number of divisors of an integer'' (including ...

τ(''n''), the

Möbius function The Möbius function is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and most oft ...

''μ''(''n''), the number of distinct prime divisors ω(''n'') of ''n'', the number of prime divisors Ω(''n'') of ''n'', the

indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...

of the

squarefree integer In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, is square- ...

s, and many other important functions in number theory, are functions of the prime signature of ''n''. In particular, τ(''n'') equals the product of the incremented by 1 exponents from the prime signature of ''n''. For example, 20 has prime signature and so the number of divisors is (2+1) × (1+1) = 6. Indeed, there are six divisors: 1, 2, 4, 5, 10 and 20. The smallest number of each prime signature is a product of

primorial In mathematics, and more particularly in number theory, primorial, denoted by "#", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function ...

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, ... . A number cannot divide another unless its prime signature is included in the other numbers prime signature in the

Young's lattice In mathematics, Young's lattice is a lattice that is formed by all integer partitions. It is named after Alfred Young, who, in a series of papers ''On quantitative substitutional analysis,'' developed the representation theory of the symmetric gr ...

Numbers with same prime signature

Sequences defined by their prime signature

Given a number with prime signature ''S'', it is * A

if ''S'' = , * A

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...

if gcd ''S'' is

even Even may refer to: General * Even (given name), a Norwegian male personal name * Even (surname) * Even (people), an ethnic group from Siberia and Russian Far East ** Even language, a language spoken by the Evens * Odd and Even, a solitaire game w ...

, * A

cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...

if gcd ''S'' is divisible by 3, * A

square-free integer In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, is square-f ...

if max ''S'' = 1, * A

cube-free integer In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, is square-fr ...

if max ''S'' ≤ 2, * A

powerful number A powerful number is a positive integer ''m'' such that for every prime number ''p'' dividing ''m'', ''p''2 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'' = ''a ...

if min ''S'' ≥ 2, * A

perfect power In mathematics, a perfect power is a natural number that is a product of equal natural factors, or, in other words, an integer that can be expressed as a square or a higher integer power of another integer greater than one. More formally, ''n'' ...

if gcd ''S'' > 1, * An

Achilles number An Achilles number is a number that is powerful but not a perfect power. A positive integer is a powerful number if, for every prime factor of , is also a divisor. In other words, every prime factor appears at least squared in the factoriza ...

if min ''S'' ≥ 2 and gcd ''S'' = 1, * ''k''-

almost prime In number theory, a natural number is called ''k''-almost prime if it has ''k'' prime factors. More formally, a number ''n'' is ''k''-almost prime if and only if Ω(''n'') = ''k'', where Ω(''n'') is the total number of primes in the prime facto ...

if sum ''S'' = ''k''.

References

{{reflist

External links

List of the first 400 prime signatures
Number theory Prime numbers

Properties

Numbers with same prime signature

Sequences defined by their prime signature

See also

References

External links