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Unitary divisor

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(Redirected from Bi-unitary divisor) A certain type of divisor of an integer

In mathematics, a natural number a is a unitary divisor (or Hall divisor) of a number b if a is a divisor of b and if a and b a {\displaystyle {\frac {b}{a}}} are coprime, having no common factor other than 1. Equivalently, a divisor a of b is a unitary divisor if and only if every prime factor of a has the same multiplicity in a as it has in b.

The concept of a unitary divisor originates from R. Vaidyanathaswamy (1931), who used the term block divisor.

Example

The integer 5 is a unitary divisor of 60, because 5 and 60 5 = 12 {\displaystyle {\frac {60}{5}}=12} have only 1 as a common factor.

On the contrary, 6 is a divisor but not a unitary divisor of 60, as 6 and 60 6 = 10 {\displaystyle {\frac {60}{6}}=10} have a common factor other than 1, namely 2.

Sum of unitary divisors

The sum-of-unitary-divisors function is denoted by the lowercase Greek letter sigma thus: σ*(n). The sum of the k-th powers of the unitary divisors is denoted by σ*k(n):

σ k ∗ ( n ) = ∑ d ∣ n gcd ( d , n / d ) = 1 d k . {\displaystyle \sigma _{k}^{*}(n)=\sum _{d\,\mid \,n \atop \gcd(d,\,n/d)=1}\!\!d^{k}.}

It is a multiplicative function. If the proper unitary divisors of a given number add up to that number, then that number is called a unitary perfect number.

Properties

Number 1 is a unitary divisor of every natural number.

The number of unitary divisors of a number n is 2, where k is the number of distinct prime factors of n. This is because each integer N > 1 is the product of positive powers p of distinct prime numbers p. Thus every unitary divisor of N is the product, over a given subset S of the prime divisors {p} of N, of the prime powers p for p ∈ S. If there are k prime factors, then there are exactly 2 subsets S, and the statement follows.

The sum of the unitary divisors of n is odd if n is a power of 2 (including 1), and even otherwise.

Both the count and the sum of the unitary divisors of n are multiplicative functions of n that are not completely multiplicative. The Dirichlet generating function is

ζ ( s ) ζ ( s − k ) ζ ( 2 s − k ) = ∑ n ≥ 1 σ k ∗ ( n ) n s . {\displaystyle {\frac {\zeta (s)\zeta (s-k)}{\zeta (2s-k)}}=\sum _{n\geq 1}{\frac {\sigma _{k}^{*}(n)}{n^{s}}}.}

Every divisor of n is unitary if and only if n is square-free.

The set of all unitary divisors of n forms a Boolean algebra with meet given by the greatest common divisor and join by the least common multiple. Equivalently, the set of unitary divisors of n forms a Boolean ring, where the addition and multiplication are given by

a ⊕ b = a b ( a , b ) 2 , a ⊙ b = ( a , b ) {\displaystyle a\oplus b={\frac {ab}{(a,b)^{2}}},\qquad a\odot b=(a,b)}

where ( a , b ) {\displaystyle (a,b)} denotes the greatest common divisor of a and b.

Odd unitary divisors

The sum of the k-th powers of the odd unitary divisors is

σ k ( o ) ∗ ( n ) = ∑ d ∣ n d ≡ 1 ( mod 2 ) gcd ( d , n / d ) = 1 d k . {\displaystyle \sigma _{k}^{(o)*}(n)=\sum _{{d\,\mid \,n \atop d\equiv 1{\pmod {2}}} \atop \gcd(d,n/d)=1}\!\!d^{k}.}

It is also multiplicative, with Dirichlet generating function

ζ ( s ) ζ ( s − k ) ( 1 − 2 k − s ) ζ ( 2 s − k ) ( 1 − 2 k − 2 s ) = ∑ n ≥ 1 σ k ( o ) ∗ ( n ) n s . {\displaystyle {\frac {\zeta (s)\zeta (s-k)(1-2^{k-s})}{\zeta (2s-k)(1-2^{k-2s})}}=\sum _{n\geq 1}{\frac {\sigma _{k}^{(o)*}(n)}{n^{s}}}.}

Bi-unitary divisors

A divisor d of n is a bi-unitary divisor if the greatest common unitary divisor of d and n/d is 1. This concept originates from D. Suryanarayana (1972). .

The number of bi-unitary divisors of n is a multiplicative function of n with average order A log ⁡ x {\displaystyle A\log x} where

A = ∏ p ( 1 − p − 1 p 2 ( p + 1 ) )   . {\displaystyle A=\prod _{p}\left({1-{\frac {p-1}{p^{2}(p+1)}}}\right)\ .}

A bi-unitary perfect number is one equal to the sum of its bi-unitary aliquot divisors. The only such numbers are 6, 60 and 90.

OEIS sequences


References

  1. R. Vaidyanathaswamy (1931). "The theory of multiplicative arithmetic functions". Transactions of the American Mathematical Society. 33 (2): 579–662. doi:10.1090/S0002-9947-1931-1501607-1.
  2. Conway, J.H.; Norton, S.P. (1979). "Monstrous Moonshine". Bulletin of the London Mathematical Society. 11 (3): 308–339.
  3. Ivić (1985) p.395
  4. Sandor et al (2006) p.115

External links


Divisibility-based sets of integers
Overview Divisibility of 60
Factorization forms
Constrained divisor sums
With many divisors
Aliquot sequence-related
Base-dependent
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