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 [math]\displaystyle{ \frac{b}{a} }[/math] 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

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

On the contrary, 6 is a divisor but not a unitary divisor of 60, as 6 and [math]\displaystyle{ \frac{60}{6}=10 }[/math] 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):

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

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

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

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

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

It is also multiplicative, with Dirichlet generating function

[math]\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}. }[/math]

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 [math]\displaystyle{ A \log x }[/math] where

[math]\displaystyle{ A = \prod_p\left({1 - \frac{p-1}{p^2(p+1)} }\right) \ . }[/math]

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

• 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 [math]\displaystyle{ \sum_{i=1}^{n}\sigma_{1}(i) }[/math]

References

• 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. Bibcode: 1990MaCom..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". http://oeis.org/A007947/a007947.pdf. 
• 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. 
• Toth, L. (2009). "On the bi-unitary analogues of Euler's arithmetical function and the gcd-sum function". J. Int. Seq. 12. https://cs.uwaterloo.ca/journals/JIS/VOL12/Toth2/toth5.html. 

External links

• Weisstein, Eric W.. "Unitary Divisor". http://mathworld.wolfram.com/UnitaryDivisor.html. 
• Mathoverflow | Boolean ring of unitary divisors