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

The integer 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}.

It is a multiplicative function. 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.

The set of all unitary divisors of n forms a Boolean algebra with meet given by the greatest common divisor and join by the least common multiple. Equivalently, the set of unitary divisors of n forms a Boolean ring, where the addition and multiplication are given by

a\oplus b={\frac {ab}{(a,b)^{2}}},\qquad a\odot b=(a,b)

where $(a,b)$ denotes the greatest common divisor of a and b. ^[2]

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^[3]

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.^[4]

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: A034444 is $2^{\omega }(n)$ , the number of unitary divisors
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.
^ Conway, J.H.; Norton, S.P. (1979). "Monstrous Moonshine". Bulletin of the London Mathematical Society. 11 (3): 308–339.
^ 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): 13–23. doi:10.2140/pjm.1959.9.13. MR 0109806.
Cohen, Eckford (1960). "Arithmetical functions associated with the unitary divisors of an integer". Mathematische Zeitschrift. 74: 66–80. doi:10.1007/BF01180473. MR 0112861. S2CID 53004302.
Cohen, Eckford (1960). "The number of unitary divisors of an integer". American Mathematical Monthly. 67 (9): 879–880. doi:10.2307/2309455. JSTOR 2309455. MR 0122790.
Cohen, Graeme L. (1990). "On an integers' infinitary divisors". Math. Comp. 54 (189): 395–411. Bibcode:1990MaCom..54..395C. doi:10.1090/S0025-5718-1990-0993927-5. MR 0993927.
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" (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.
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. Zbl 1151.11300.
Toth, L. (2009). "On the bi-unitary analogues of Euler's arithmetical function and the gcd-sum function". J. Int. Seq. 12.

External links

[1] 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.

[2] Conway, J.H.; Norton, S.P. (1979). "Monstrous Moonshine". Bulletin of the London Mathematical Society. 11 (3): 308–339.

[Ivic395-3] Ivić (1985) p.395

[HNTI115-4] Sandor et al (2006) p.115

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Divisibility-based sets of integers
Overview	Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic
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 Descartes Erdős–Nicolas
With many divisors	Abundant Primitive abundant Highly abundant Superabundant Colossally abundant Highly composite Superior highly composite Weird
Aliquot sequence-related	Untouchable Amicable (Triple) Sociable Betrothed
Base-dependent	Equidigital Extravagant Frugal Harshad Polydivisible Smith
Other sets	Arithmetic Deficient Friendly Solitary Sublime Harmonic divisor Refactorable Superperfect

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