# 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 $\displaystyle{ \frac{b}{a} }$ are coprime, having no common factor other than 1. Thus, 5 is a unitary divisor of 60, because 5 and $\displaystyle{ \frac{60}{5}=12 }$ have only 1 as a common factor, while 6 is a divisor but not a unitary divisor of 60, as 6 and $\displaystyle{ \frac{60}{6}=10 }$ have a common factor other than 1, namely 2. 1 is a unitary divisor of every natural number. 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 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):

$\displaystyle{ \sigma_k^*(n) = \sum_{d \,\mid\, n \atop \gcd(d,\,n/d)=1} \!\! d^k. }$

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

The concept of a unitary divisor originates from R. Vaidyanathaswamy (1931) [The theory of multiplicative arithmetic functions. Transactions of the American Mathematical Society, 33(2), 579--662] who used the term block divisor.

## Properties

The number of unitary divisors of a number n is 2k, where k is the number of distinct prime factors of n.

This is because each integer N > 1 is the product of positive powers prp 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 prp for pS. If there are k prime factors, then there are exactly 2k 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

$\displaystyle{ \frac{\zeta(s)\zeta(s-k)}{\zeta(2s-k)} = \sum_{n\ge 1}\frac{\sigma_k^*(n)}{n^s}. }$

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

$\displaystyle{ \sigma_k^{(o)*}(n) = \sum_{{d \,\mid\, n \atop d \equiv 1 \pmod 2} \atop \gcd(d,n/d)=1} \!\! d^k. }$

It is also multiplicative, with Dirichlet generating function

$\displaystyle{ \frac{\zeta(s)\zeta(s-k)(1-2^{k-s})}{\zeta(2s-k)(1-2^{k-2s})} = \sum_{n\ge 1}\frac{\sigma_k^{(o)*}(n)}{n^s}. }$

## 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 $\displaystyle{ A \log x }$ where[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.[2]

## OEIS sequences

• is σ*0(n
•   is σ*1(n
•   to are σ*2(n) to σ*8(n
•   is σ(o)*0(n
•   is σ(o)*1(n
•   is $\displaystyle{ \sum_{i=1}^{n}\sigma_{1}(i) }$

## References

1. Ivić (1985) p.395
2. Sandor et al (2006) p.115
• Richard K. Guy (2004). Unsolved Problems in Number Theory. Springer-Verlag. p. 84. ISBN 0-387-20860-7.  Section B3.
• Paulo Ribenboim (2000). My Numbers, My Friends: Popular Lectures on Number Theory. Springer-Verlag. p. 352. ISBN 0-387-98911-0.
• Cohen, Eckford (1959). "A class of residue systems (mod r) and related arithmetical functions. I. A generalization of Möbius inversion". Pacific J. Math. 9 (1): 13–23. doi:10.2140/pjm.1959.9.13.
• Cohen, Eckford (1960). "Arithmetical functions associated with the unitary divisors of an integer". Mathematische Zeitschrift 74: 66–80. doi:10.1007/BF01180473.
• Cohen, Eckford (1960). "The number of unitary divisors of an integer". American Mathematical Monthly 67 (9): 879–880. doi:10.2307/2309455.
• Cohen, Graeme L. (1990). "On an integers' infinitary divisors". Math. Comp. 54 (189): 395–411. doi:10.1090/S0025-5718-1990-0993927-5. Bibcode1990MaCom..54..395C.
• Cohen, Graeme L. (1993). "Arithmetic functions associated with infinitary divisors of an integer". Int. J. Math. Math. Sci. 16 (2): 373–383. doi:10.1155/S0161171293000456.
• Finch, Steven (2004). "Unitarism and Infinitarism".
• Ivić, Aleksandar (1985). The Riemann zeta-function. The theory of the Riemann zeta-function with applications. A Wiley-Interscience Publication. New York etc.: John Wiley & Sons. p. 395. ISBN 0-471-80634-X.
• Mathar, R. J. (2011). "Survey of Dirichlet series of multiplicative arithmetic functions". arXiv:1106.4038 [math.NT]. Section 4.2
• Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds (2006). Handbook of number theory I. Dordrecht: Springer-Verlag. ISBN 1-4020-4215-9.
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