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\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. 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" (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. 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 Composite Semiprime Pronic Sphenic Square-free Powerful Perfect power Achilles Smooth Regular Rough Unusual

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

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

Aliquot sequence-related	Untouchable Amicable Sociable Betrothed

Other sets	Deficient Friendly Solitary Sublime Harmonic divisor Frugal Equidigital Extravagant

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