Unitary divisor

In mathematics, a natural number a is a unitary 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. Thus, 5 is a unitary divisor of 60, because 5 and $\frac{60}{5}=12$ have only 1 as a common factor, while 6 is a divisor but not a unitary divisor of 60, as 6 and $\frac{60}{6}=10$ have a common factor other than 1, namely 2. 1 is a unitary divisor of every natural number.

Equivalently, a given 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):

\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

The number of unitary divisors of a number n is 2^k, where k is the number of distinct prime factors of n. 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\ge 1}\frac{\sigma_k^*(n)}{n^s}.

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\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. The number of bi-unitary divisors of n is a multiplicative function of n with average order $A \log x$ where^[1]

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]

References

↑ Ivić (1985) p.395
↑ 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). pp. 13—23. MR 0109806.
Cohen, Eckford (1960). "Arithmetical functions associated with the unitary divisors of an integer". Mathematische Zeitschrift 74. pp. 66—80. doi:10.1007/BF01180473. MR 0112861.
Cohen, Eckford (1960). "The number of unitary divisors of an integer". American mathematical monthly 67 (9). pp. 879—880. MR 0122790.
Cohen, Graeme L. (1990). "On an integers' infinitary divisors". Math. Comp. 54 (189). pp. 395—411. doi:10.1090/S0025-5718-1990-0993927-5. MR 0993927.
Cohen, Graeme L. (1993). "Arithmetic functions associated with infinitary divisors of an integer". Intl. J. Math. Math. Sci. 16 (2). pp. 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. Zbl 0556.10026.
Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006). Handbook of number theory I. Dordrecht: Springer-Verlag. ISBN 1-4020-4215-9. Zbl 1151.11300.

External links

Weisstein, Eric W., "Unitary Divisor", MathWorld.

OEIS sequences

A034444 is σ₀(n)
A034448 is σ₁(n)
A034676 to A034682 are σ₂(n) to σ₈(n)
A068068 is σ^(o)*₀(n)
A192066 is σ^(o)*₁(n)

Divisibility-based sets of integers

Overview	Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic Arithmetic number

Factorization forms	Prime number Composite number Semiprime number Pronic number Sphenic number Square-free number Powerful number Perfect power Achilles number Smooth number Regular number Rough number Unusual number

Constrained divisor sums	Perfect number Almost perfect number Quasiperfect number Multiply perfect number Hemiperfect number Hyperperfect number Superperfect number Unitary perfect number Semiperfect number Practical number Erdős–Nicolas number

With many divisors	Abundant number Primitive abundant number Highly abundant number Superabundant number Colossally abundant number Highly composite number Superior highly composite number Weird number

Aliquot sequence-related	Untouchable number Amicable number Sociable number Betrothed number

Other sets	Deficient number Friendly number Solitary number Sublime number Harmonic divisor number Frugal number Equidigital number Extravagant number