unitary divisor

The integer 5 is a unitary divisor of 60, because 5 and $\backslash frac\{60\}\{5\}=12$ have only 1 as a common factor.

On the contrary, 6 is a divisor but not a unitary divisor of 60, as 6 and $\backslash frac\{60\}\{6\}=10$ have a common factor other than 1, namely 2.

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):

:$\backslash sigma\_k^*(n)\; =\; \backslash sum\_\{d\; \backslash ,\backslash mid\backslash ,\; n\; \backslash atop\; \backslash gcd(d,\backslash ,n/d)=1\}\; \backslash !\backslash !\; d^k.$

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.

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

:$\backslash frac\{\backslash zeta(s)\backslash zeta(s-k)\}\{\backslash zeta(2s-k)\}\; =\; \backslash sum\_\{n\backslash ge\; 1\}\backslash frac\{\backslash sigma\_k^*(n)\}\{n^s\}.$

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

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

where $(a,b)$ denotes the greatest common divisor of a and b. {{cite journal |last1=Conway |first1=J.H. |last2=Norton |first2=S.P. |title=Monstrous Moonshine |journal=Bulletin of the London Mathematical Society |volume=11 |issue=3 |pages=308-339 |year=1979 | url= https://doi.org/10.1112/blms/11.3.308 }}

The sum of the k-th powers of the odd unitary divisors is

:$\backslash sigma\_k^\{(o)*\}(n)\; =\; \backslash sum\_\{\{d\; \backslash ,\backslash mid\backslash ,\; n\; \backslash atop\; d\; \backslash equiv\; 1\; \backslash pmod\; 2\}\; \backslash atop\; \backslash gcd(d,n/d)=1\}\; \backslash !\backslash !\; d^k.$

It is also multiplicative, with Dirichlet generating function

:$\backslash frac\{\backslash zeta(s)\backslash zeta(s-k)(1-2^\{k-s\})\}\{\backslash zeta(2s-k)(1-2^\{k-2s\})\}\; =\; \backslash sum\_\{n\backslash ge\; 1\}\backslash frac\{\backslash sigma\_k^\{(o)*\}(n)\}\{n^s\}.$

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 $A\; \backslash log\; x$ whereIvić (1985) p.395

:$A\; =\; \backslash prod\_p\backslash left(\{1\; -\; \backslash frac\{p-1\}\{p^2(p+1)\}\; \}\backslash right)\; \backslash \; .$

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.Sandor et al (2006) p.115

• {{OEIS2C|A034444}} is σ^*₀(n)
• {{OEIS2C|A034448}} is σ^*₁(n) 
• {{OEIS2C|A034676}} to {{OEIS2C|A034682}} are σ^*₂(n) to σ^*₈(n) 
• {{OEIS2C|A034444}} is $2^\backslash omega(n)$, the number of unitary divisors
• {{OEIS2C|A068068}} is σ^(o)*₀(n) 
• {{OEIS2C|A192066}} is σ^(o)*₁(n) 
• {{OEIS2C|A064609}} is $\backslash sum\_\{i=1\}^\{n\}\backslash sigma\_\{1\}(i)$
• {{OEIS2C|A306071}} is $A$

{{reflist}}

• {{cite book|author=Richard K. Guy|authorlink=Richard K. Guy|title=Unsolved Problems in Number Theory|publisher=Springer-Verlag|year=2004|isbn=0-387-20860-7 | page=84}} Section B3.
• {{cite book | title=My Numbers, My Friends: Popular Lectures on Number Theory | author=Paulo Ribenboim | authorlink=Paulo Ribenboim | publisher=Springer-Verlag | year=2000 | isbn=0-387-98911-0 | page=352 }}
• {{cite journal|first1=Eckford|last1=Cohen|title=A class of residue systems (mod r) and related arithmetical functions. I. A generalization of Möbius inversion|journal=Pacific J. Math.|volume=9|number=1 |pages=13–23|year=1959|mr=0109806|doi=10.2140/pjm.1959.9.13|doi-access=free}}
• {{cite journal|first1=Eckford|last1=Cohen|title=Arithmetical functions associated with the unitary divisors of an integer|journal=Mathematische Zeitschrift|volume=74|year=1960|pages=66–80|mr=0112861 |doi=10.1007/BF01180473|s2cid=53004302}}
• {{cite journal|first1=Eckford|last1=Cohen|title=The number of unitary divisors of an integer|volume=67|number=9|pages=879–880|mr=0122790|year=1960|journal=American Mathematical Monthly|doi=10.2307/2309455|jstor=2309455}}
• {{cite journal|first1=Graeme L.|last1=Cohen|title=On an integers' infinitary divisors|volume=54|number=189 |pages=395–411|mr=0993927|doi=10.1090/S0025-5718-1990-0993927-5|journal=Math. Comp.|year=1990|bibcode=1990MaCom..54..395C|doi-access=free}}
• {{cite journal|first1=Graeme L.|last1=Cohen|title=Arithmetic functions associated with infinitary divisors of an integer|volume=16|number=2|pages=373–383|doi=10.1155/S0161171293000456|journal=Int. J. Math. Math. Sci.|year=1993|doi-access=free}}
• {{cite web|first1=Steven|last1=Finch|title=Unitarism and Infinitarism|url=http://oeis.org/A007947/a007947.pdf|year=2004}}
• {{cite book | last=Ivić | first=Aleksandar | title=The Riemann zeta-function. The theory of the Riemann zeta-function with applications | series=A Wiley-Interscience Publication | location=New York etc. | publisher=John Wiley & Sons | year=1985 | isbn=0-471-80634-X | zbl=0556.10026 | page=395 }}
• {{cite arXiv|eprint=1106.4038 | first1=R. J.|last1=Mathar|year=2011|title=Survey of Dirichlet series of multiplicative arithmetic functions| class=math.NT}} Section 4.2
• {{cite book | editor1-last=Sándor | editor1-first=József | editor2-last=Mitrinović | editor2-first=Dragoslav S. | editor3-last=Crstici |editor3-first=Borislav | title=Handbook of number theory I | location=Dordrecht | publisher=Springer-Verlag | year=2006 | isbn=1-4020-4215-9 | zbl=1151.11300 }}
• {{cite journal| first1=L. | last1=Toth|title=On the bi-unitary analogues of Euler's arithmetical function and the gcd-sum function| url=https://cs.uwaterloo.ca/journals/JIS/VOL12/Toth2/toth5.html | year=2009 | journal=J. Int. Seq. | volume=12}}

• {{MathWorld |urlname=UnitaryDivisor |title=Unitary Divisor}}
• [https://mathoverflow.net/questions/369751/boolean-ring-of-unitary-divisors-structure-of-unitary-divisors Mathoverflow | Boolean ring of unitary divisors]

{{Divisor classes}}

Category:Number theory