Unitary divisor

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 ${\frac {b}{a}}$ 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),^[1] who used the term block divisor.

Example

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

On the contrary, 6 is a divisor but not a unitary divisor of 60, as 6 and ${\frac {60}{6}}=10$ 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):

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

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^k, 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^r_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^r_p for p ∈ S. If there are k prime factors, then there are exactly 2^k 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

{\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.

Odd unitary divisors

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

\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

{\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 an integer, in The Theory of Arithmetic Functions, Lecture Notes in Mathematics 251: 273–282, New York, Springer–Verlag].

The number of bi-unitary divisors of n is a multiplicative function of n with average order $A\log x$ where^[2]

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.^[3]

OEIS sequences

OEIS: A034444 is σ^*₀(n)
OEIS: A034448 is σ^*₁(n)
OEIS: A034676 to OEIS: A034682 are σ^*₂(n) to σ^*₈(n)
OEIS: A068068 is σ^(o)*₀(n)
OEIS: A192066 is σ^(o)*₁(n)
OEIS: A064609 is $\sum _{i=1}^{n}\sigma _{1}(i)$
OEIS: A306071 is $A$

References

^ 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.
^ Ivić (1985) p.395
^ Sandor et al (2006) p.115

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